MAXIMUM AND MINIMUM VALUES 4 APPLICATIONS OF DIFFERENTIATION 4.1 MAXIMUM AND MINIMUM VALUES

Maximum and Minimum Values Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) way of doing something.

Maximum and Minimum Values Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) way of doing something. ▪ What is the shape of a can that minimizes manufacturing costs? ▪ What is the maximum acceleration of a space shuttle?

Maximum and Minimum Values Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) way of doing something. ▪ What is the shape of a can that minimizes manufacturing costs? ▪ What is the maximum acceleration of a space shuttle? These problems can be reduced to finding the maximum or minimum values of a function.

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values.

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5).

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5).

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 3

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 In other words, the largest value of f is f (3) = 5. 3

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 In other words, the largest value of f is f (3) = 5. 3

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 In other words, the largest value of f is f (3) = 5. 2 3 6

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 Likewise, the smallest value is f (6) = 2. In other words, the largest value of f is f (3) = 5. 2 3 6

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 Likewise, the smallest value is f (6) = 2. In other words, the largest value of f is f (3) = 5. 3

Maximum and Minimum Values Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point (3, 5). 5 Likewise, the smallest value is f (6) = 2. In other words, the largest value of f is f (3) = 5. 3 We say that f (3) = 5 is the absolute maximum of f and f (6) = 2 is the absolute minimum of f.

Maximum and Minimum Values One World Trade Center Empire State Bldg 432 Park Avenue

Maximum and Minimum Values In general, we use the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. Then 𝑓(𝑐) is the absolute maximum value of 𝑓 on 𝐷 if 𝑓(𝑐)≥𝑓(𝑥) for all 𝑥 in 𝐷. absolute minimum value of 𝑓 on 𝐷 if 𝑓(𝑐)≤𝑓(𝑥) for all 𝑥 in 𝐷.

Maximum and Minimum Values In general, we use the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. Then 𝑓(𝑐) is the absolute maximum value of 𝑓 on 𝐷 if 𝑓(𝑐)≥𝑓(𝑥) for all 𝑥 in 𝐷. absolute minimum value of 𝑓 on 𝐷 if 𝑓(𝑐)≤𝑓(𝑥) for all 𝑥 in 𝐷. An absolute maximum or minimum is sometimes called a global maximum or minimum.

Maximum and Minimum Values In general, we use the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. Then 𝑓(𝑐) is the absolute maximum value of 𝑓 on 𝐷 if 𝑓(𝑐)≥𝑓(𝑥) for all 𝑥 in 𝐷. absolute minimum value of 𝑓 on 𝐷 if 𝑓(𝑐)≤𝑓(𝑥) for all 𝑥 in 𝐷. An absolute maximum or minimum is sometimes called a global maximum or minimum. The maximum and minimum values of f are called extreme values of f.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a.

Maximum and Minimum Values Absolute max This is the graph of a function f with absolute maximum at d and absolute minimum at a. 𝑓(𝑑)

Maximum and Minimum Values Absolute max This is the graph of a function f with absolute maximum at d and absolute minimum at a. 𝑓(𝑑)

Maximum and Minimum Values Absolute max This is the graph of a function f with absolute maximum at d and absolute minimum at a. 𝑓(𝑑) 𝑓(𝑎) Absolute min

Maximum and Minimum Values Absolute max This is the graph of a function f with absolute maximum at d and absolute minimum at a. 𝑓(𝑑) Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. 𝑓(𝑎) Absolute min

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. 𝑓(𝑏) If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f.

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f. 𝑓(𝑐) Likewise, f (c) is called a local minimum value of f because f (c)  f (x) for x near c

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f. 𝑓(𝑐) Likewise, f (c) is called a local minimum value of f because f (c)  f (x) for x near c [in the interval (b, d), for instance].

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f. Likewise, f (c) is called a local minimum value of f because f (c)  f (x) for x near c [in the interval (b, d), for instance].

Maximum and Minimum Values This is the graph of a function f with absolute maximum at d and absolute minimum at a. Note that (𝑑,𝑓 𝑑 ) is the highest point on the graph and (𝑎,𝑓 𝑎 ) is the lowest point. If we consider only values of x near b [for instance, if we only consider the interval (a, c)], then f (b) is the largest of those values of f (x) and is called a local maximum value of f. 𝑓(𝑒) Likewise, f (c) is called a local minimum value of f because f (c)  f (x) for x near c The function also has a local minimum 𝑓(𝑒) at 𝑒. [in the interval (b, d), for instance].

Maximum and Minimum Values In general, we have the following definition.

Maximum and Minimum Values In general, we have the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐.

Maximum and Minimum Values In general, we have the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐. If we say that something is true near c, we mean that it is true on some open interval containing c.

Maximum and Minimum Values In general, we have the following definition. DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. If we say that something is true near c, we mean that it is true on some open interval containing c.

Maximum and Minimum Values DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. This is the graph of

Maximum and Minimum Values DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. This is the graph of We can see that f (1) = 5 is a local maximum

Maximum and Minimum Values DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. This is the graph of We can see that f (1) = 5 is a local maximum and the absolute maximum is f (–1) = 37.

Maximum and Minimum Values DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. This is the graph of We can see that f (1) = 5 is a local maximum and the absolute maximum is f (–1) = 37. (This absolute maximum is not a local maximum because it occurs at an endpoint.)

Maximum and Minimum Values DEFINITION Let 𝑐 be a number in the domain 𝐷 of a function 𝑓. 𝑓(𝑐) is a local maximum value of 𝑓 if 𝑓(𝑐)≥𝑓(𝑥) when 𝑥 is near 𝑐*. local minimum value of 𝑓 if 𝑓(𝑐)≤𝑓(𝑥) when 𝑥 is near 𝑐*. *technically speaking, ‘near 𝑐” means ‘on an open interval including 𝑐’. This is the graph of We can see that f (1) = 5 is a local maximum and the absolute maximum is f (–1) = 37. (This absolute maximum is not a local maximum because it occurs at an endpoint.) Also, f (0) = 0 is a local minimum and f (3) = –27 is both a local and an absolute minimum

Maximum and Minimum Values We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values.

Maximum and Minimum Values We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏].

Maximum and Minimum Values We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏].

Maximum and Minimum Values We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏]. Note that an extreme value can be taken on more than once.

Maximum and Minimum Values This theorem says that a continuous function on a closed interval has a maximum value and a minimum value, BUT it does not tell us how to find these extreme values. We start by looking for local extreme values. We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏].

Maximum and Minimum Values This theorem says that a continuous function on a closed interval has a maximum value and a minimum value, BUT it does not tell us how to find these extreme values. We start by looking for local extreme values. We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏]. For example, here it appears that at the maximum & minimum points, tangents are horizontal (slope is 0).

Maximum and Minimum Values We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values. This theorem says that a continuous function on a closed interval has a maximum value and a minimum value, BUT it does not tell us how to find these extreme values. We start by looking for local extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏]. For example, here it appears that at the maximum & minimum points, tangents are horizontal (slope is 0). So it appears that f (c) = 0 and f (d) = 0.

Maximum and Minimum Values This theorem says that a continuous function on a closed interval has a maximum value and a minimum value, BUT it does not tell us how to find these extreme values. We start by looking for local extreme values. THE EXTREME VALUE THEOREM If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 attains an absolute maximum value, 𝑓(𝑐) and an absolute minimum value, 𝑓(𝑑) at some numbers 𝑐 and 𝑑 in [𝑎,𝑏]. For example, here it appears that at the maximum & minimum points, tangents are horizontal (slope is 0). So it appears that f (c) = 0 and f (d) = 0. FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Fraction is 0 only if NUM=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Fraction is 0 only if NUM=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Fraction is 0 only if NUM=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Doesn’t exist when dividing by 0. 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0 which occurs if 𝑥=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Doesn’t exist when dividing by 0. 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0 which occurs if 𝑥=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Doesn’t exist when dividing by 0. 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0 which occurs if 𝑥=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of Doesn’t exist when dividing by 0. 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0 which occurs if 𝑥=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. Find the critical numbers of So the critical numbers are 3 2 and 0. Doesn’t exist when dividing by 0. 𝑓 ′ (𝑥) does not exist when 5 𝑥 2 5 =0 which occurs if 𝑥=0.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. This definition allows us to rephrase Fermat’s theorem as follows:

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. This definition allows us to rephrase Fermat’s theorem as follows: If 𝑓 has a local maximum or minimum at 𝑐, then 𝑐 is a critical number of 𝑓.

Maximum and Minimum Values FERMAT’S THEOREM If 𝑓 has a local maximum or minimum at 𝑐, and if 𝑓′(𝑐) exists, then 𝑓 ′ 𝑐 =0. Now we will define something called a critical number. DEFINTION A critical number of a function 𝑓 is a number 𝑐 in the DOMAIN of 𝑓 such that either 𝑓 ′ 𝑐 =0 or 𝑓 ′ 𝑐 does not exist. This definition allows us to rephrase Fermat’s theorem as follows: If 𝑓 has a local maximum or minimum at 𝑐, then 𝑐 is a critical number of 𝑓. So, to find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local (in which case it occurs at a critical number) or it occurs at an endpoint of the interval.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function continuous

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function continuous closed interval

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function Note that both are in the closed interval , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. The function values at the endpoints of the closed interval are

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function Largest value is absolute max 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function Largest value is absolute max 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are Absolute maximum value is 17.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are Absolute maximum value is 17.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function Smallest value is absolute min 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are Absolute maximum value is 17.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function Smallest value is absolute min 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are Absolute maximum value is 17. Absolute minimum value is −3.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values of the function 𝑓 0 =1 𝑓 2 =−3 , critical numbers are 𝑥=0 or 𝑥=2. 𝑓 − 1 2 = 1 8 𝑓 4 =17 The function values at the endpoints of the closed interval are Absolute maximum value is 17. Absolute minimum value is −3.

Maximum and Minimum Values All of this leads to the following 3-step procedure which always works. Find the absolute maximum and minimum values Absolute maximum value is 17. Absolute minimum value is −3.

Focus on Homework

Focus on Homework

Focus on Homework 5 4 3 2

Focus on Homework 5 5 4 3 2

Focus on Homework 5 5 4 3 2

Focus on Homework 5 5 4 3 2 b/c this is open, there is no abs min value

Focus on Homework 5 5 4 DNE 3 2 b/c this is open, there is no abs min value

Focus on Homework 5 5 4 DNE 3 2

Focus on Homework b/c this is open, there is no local max value here 5 5 4 DNE 3 2

Focus on Homework b/c this is open, there is no local max value here 5 5 4 DNE 3 2 But there is a local max value here

Focus on Homework b/c this is open, there is no local max value here 5 5 4 DNE 3 2 4, 5 But there is a local max value here

Focus on Homework 5 5 4 DNE 3 2 4, 5

Focus on Homework 5 5 4 DNE 3 2 4, 5

Focus on Homework 5 5 4 DNE 3 2 4, 5 2, 3

Focus on Homework 5 5 4 DNE 3 2 4, 5 2, 3

Focus on Homework

Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 .

Focus on Homework 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥

Focus on Homework 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units.

Focus on Homework 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units.

Focus on Homework 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 )

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

…or you could just use the calculator to get points to plot. Focus on Homework One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) …or you could just use the calculator to get points to plot.

Focus on Homework ln 20 𝐷𝑁𝐸 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 (4, ln 4 ) Local extrema cannot occur at an endpoint. …or you could just use the calculator to get points to plot.

Focus on Homework ln 20 𝐷𝑁𝐸 𝐷𝑁𝐸 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 𝐷𝑁𝐸 (4, ln 4 ) Local extrema cannot occur at an endpoint. …or you could just use the calculator to get points to plot.

Focus on Homework ln 20 𝐷𝑁𝐸 𝐷𝑁𝐸 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 𝐷𝑁𝐸 (4, ln 4 ) Local extrema cannot occur at an endpoint. …or you could just use the calculator to get points to plot.

Focus on Homework ln 20 𝐷𝑁𝐸 𝐷𝑁𝐸 𝐷𝑁𝐸 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 One way to get a quick sketch is by rewriting the function using: log 𝑏 𝑢𝑣 = log 𝑏 𝑢 + log 𝑏 𝑣 . 𝑓 𝑥 = 𝑙𝑛 5𝑥 = 𝑙𝑛 5 + 𝑙𝑛 𝑥 So, the graph of 𝑓 𝑥 = 𝑙𝑛 5𝑥 is the graph of 𝑓 𝑥 = 𝑙𝑛 𝑥 shifted up 𝑙𝑛 5 units. ln 20 𝐷𝑁𝐸 𝑦=ln⁡(5𝑥) (4, ln 20 ) ln 5 𝐷𝑁𝐸 (4, ln 4 ) Local extrema cannot occur at an endpoint. 𝐷𝑁𝐸 …or you could just use the calculator to get points to plot.

Focus on Homework

Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 )

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0

Focus on Homework 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 𝑥=0

No + number raised to a power can = 0 Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 No + number raised to a power can = 0 𝑥=0

No + number raised to a power can = 0 Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 No + number raised to a power can = 0 𝑥= 5 8 𝑥=0

No + number raised to a power can = 0 Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 No + number raised to a power can = 0 𝑥= 5 8 𝑥=0

𝒇 ′ 𝒙 has no undefined values. No + number raised to a power can = 0 Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝒇 ′ 𝒙 has no undefined values. 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 No + number raised to a power can = 0 𝑥= 5 8 𝑥=0

𝒇 ′ 𝒙 has no undefined values. No + number raised to a power can = 0 Focus on Homework CRITICAL NUMBERS Values in the domain of 𝑓 for which: 𝑓 ′ 𝑥 =0 or 𝑓 ′ 𝑥 does not exist 0 , 5 8 𝑓 ′ 𝑥 = 𝑥 5 𝑒 −8𝑥 (−8) + 𝑒 −8𝑥 (5 𝑥 4 ) 𝑓 ′ 𝑥 =−8 𝑥 5 𝑒 −8𝑥 +5 𝑥 4 𝑒 −8𝑥 𝑓 ′ 𝑥 = 𝑥 4 𝑒 −8𝑥 −8𝑥+5 =0 𝒇 ′ 𝒙 has no undefined values. 𝑥 4 =0 or 𝑒 −8𝑥 =0 or −8𝑥+5=0 No + number raised to a power can = 0 𝑥= 5 8 𝑥=0