The mileage of a certain car can be approximated by: At what speed should you drive the car to obtain the best gas mileage? Of course, this problem isn’t entirely realistic, since it is unlikely that you would have an equation like this for your car.

We could solve the problem graphically:

We could solve the problem graphically: On the TI-84, we use 2nd CALC 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

We could solve the problem graphically: The car will get approximately 32 miles per gallon when driven at 38.6 miles per hour. On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

Notice that at the top of the curve, the horizontal tangent has a slope of zero. Traditionally, this fact has been used both as an aid to graphing by hand and as a method to find maximum (and minimum) values of functions.

Slope of the Tangent Line f ’(a) is the slope of the tangent line to the graph of f at x = a. This tangent slope interpretation of f has several useful geometric implications. The values of the derivative enables us to detect when a function is increasing or decreasing.

If f ’(x) > 0 on an interval (a, b) then f is increasing on that interval. If f ’(x) < 0 on an interval (a, b) then f is decreasing on that interval.

Point (c, f(c)) is called a critical point of the graph of f. A number c in the interior of the domain of a function f is called a critical number if either: f ’(c) = 0 f ’(c) does not exist or c c Point (c, f(c)) is called a critical point of the graph of f.

The only inputs, x = c, at which the derivative of a function can change sign (+ → – or – → +) are where f ‘(c) is at the critical numbers of: 0 or undefined

A local maximum (or relative maximum) for a function f occurs at a point x = c if f(c) is the largest value of f in some interval (a < c < b) centered at x = c. absolute max. local max. A global maximum (or absolute maximum) for a function f occurs at a point x = c if f(c) is the largest value of f for every x in the domain.

A local minimum (or relative minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f in some interval (a < c < b) centered at x = c. local min. absolute min. A global minimum (or absolute minimum) for a function f occurs at a point x = c if f(c) is the smallest value of f for every x in the domain.

A function can have at most one global max. and one global min A function can have at most one global max. and one global min., though this value can be assumed at many points. Example: f(x) = sin x has a global max. value of 1 at many inputs x. Maximum and minimum values of a function are collectively referred to as extreme values (or extrema). A critical point with zero derivative but no maximum or min. is called a plateau point.

Example 1: Find critical numbers for the given Algebraically, Remember, critical numbers occur if the derivative does not exist or is zero. Our critical numbers!!

Example 1: Graphically, First check the values of the derivatives A number-line graph for f and f ’ helps. It is a convenient way to sketch a graph. Graphically, Remember, if f ’<0, then f(x) is decreasing and if f ’(x)>0 f(x) is increasing. First check the values of the derivatives before and after your critical #'s. f '(x) – + – + x –1 1

Example 1: Graphically, Second identify the local mins. and maxs. on Recall, local min. at x=c if f(c) is the smallest value in the interval. Graphically, Recall, local max. at x=c if f(c) is the largest value in the interval. max. f (x) Second identify the local mins. and maxs. on your interval. min. min. f '(x) – + – + x –1 1

Extreme values can be in the interior or the end points of a function. No Absolute Maximum Absolute Minimum

Absolute Maximum Absolute Minimum

Absolute Maximum No Minimum

No Maximum No Minimum

Extreme Value Theorem: If f is continuous over a closed interval, then f has a maximum and minimum value over that interval. Maximum & minimum at interior points Maximum & minimum at endpoints Maximum at interior point, minimum at endpoint

Local Extreme Values: A local maximum is the maximum value within some open interval. A local minimum is the minimum value within some open interval.

Absolute maximum (also local maximum) Local maximum Local minimum Local minimum Local extremes are also called relative extremes. Absolute minimum (also local minimum)

Absolute maximum (also local maximum) Local maximum Local minimum Notice that local extremes in the interior of the function occur where is zero or is undefined.

EXAMPLE FINDING ABSOLUTE EXTREMA Find the absolute maximum and minimum values of on the interval . There are no values of x that will make the first derivative equal to zero. The first derivative is undefined at x=0, so (0,0) is a critical point. Because the function is defined over a closed interval, we also must check the endpoints.

At: To determine if this critical point is actually a maximum or minimum, we try points on either side, without passing other critical points. Since 0<1, this must be at least a local minimum, and possibly a global minimum. At: At:

At: To determine if this critical point is actually a maximum or minimum, we try points on either side, without passing other critical points. Absolute minimum: maximum: Since 0<1, this must be at least a local minimum, and possibly a global minimum. At: At:

Absolute maximum (3,2.08) Absolute minimum (0,0)

Finding Maximums and Minimums Analytically: 1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2 Find the value of the function at each critical point. 3 Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums. 4 For closed intervals, check the end points as well.

Critical points are not always extremes! (not an extreme)

(not an extreme) p

Example #1 Find the critical numbers of each function

Example #2 Locate the absolute extrema of each function on the closed interval.

Example #3 Locate the absolute extrema of each function on the closed interval.

Example #4 Locate the absolute extrema of each function on the closed interval.

Example #5 Locate the absolute extrema of each function on the closed interval.

Example #6 Locate the absolute extrema of each function on the closed interval.

Example #7 Locate the absolute extrema of each function on the closed interval.

Example #8 Locate the absolute extrema of each function on the closed interval.

If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b: Mean Value Theorem for Derivatives The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope. The Mean Value Theorem only applies over a closed interval.

Tangent parallel to chord. Slope of tangent: Slope of chord:

Example #9 Find all values of c in the interval (a,b) such that

Example #10 Find all values of c in the interval (a,b) such that

Example #11 Find all values of c in the interval (a,b) such that