Maximum and Minimum Values 3.1 Maximum and Minimum Values

Maximum and Minimum values of a function Some of the most important applications of calculus are optimization problems, which find the optimal way of doing something. Meaning: find the maximum or minimum values of some function. So what are maximum and minimum values? Example: highest point on the graph of the function f shown is the point (3, 5) so the largest value of f is f (3) = 5. Likewise, the smallest value is f (6) = 2. Figure 1

Absolute Maximum and Minimum The maximum and minimum values of f are called extreme values of f.

Local Maximum and Minimum Something is true near c means that it is true on some open interval containing c.

Example 1

Example 2 Figure 2 shows the graph of a function f with several extrema: At a : absolute minimum is f(a) At b : local maximum is f(b) At c : local minimum is f(c) At d : absolute (also local) maximum is f(d) At e : local minimum is f(e) Abs min f (a), abs max f (d) loc min f (c) , f(e), loc max f (b), f (d) Figure 2

Practice! Define all local and absolute extrema of the graph below

Critical Point

Counter examples: f’ exists but there is no local min or max f’ doesn’t exist but there is a local min or max

Extreme Value Theorem The following theorem gives conditions under which a function is guaranteed to have extreme values.

Extreme Value Theorem: Absolute min and max The Extreme Value Theorem is illustrated below: Note that an extreme value can be taken on more than once. Figure 7

Extreme Value Theorem : Absolute min and max The Extreme Value 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. Graph of a function f with a local maximum at c and a local minimum at d. Figure 10

Finding Absolute Maximum and Minimum of f: To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local or it occurs at an endpoint of the interval. Thus the following three-step procedure always works.

Practice Find the absolute values of the function and where they occur (worksheet 3.1, # 5)

3.2 The Mean Value Theorem

Rolle’s Theorem

Examples: Figure 1 shows the graphs of four such functions. (a) (b) (d) Figure 1

The Mean Value Theorem This theorem is an extension of Rolle’s Theorem

Example Show that f(x) satisfies the Mean Value Theorem on [a,b] f (x) = x3 – x, Interval: a = 0, b = 2. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 2] and differentiable on (0, 2). Therefore, by the Mean Value Theorem, there is a number c in (0, 2) such that f (2) – f (0) = f (c)(2 – 0)

Example - proof f (2) = 6, f (0) = 0, and f (x) = 3x2 – 1, so this equation becomes: 6 = (3c2 – 1)2 = 6c2 – 2 which gives that is, c = But c must lie in (0, 2), so

The Mean Value Theorem The Mean Value Theorem can be used to establish some of the basic facts of differential calculus.