3 1 Maximum and Minimum Values Maximum and

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3. 1 Maximum and Minimum Values

3. 1 Maximum and Minimum Values

Maximum and Minimum values of a function Some of the most important applications of

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 2

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

Absolute Maximum and Minimum The maximum and minimum values of f are called extreme values of f. 3

Local Maximum and Minimum Something is true near c means that it is true

Local Maximum and Minimum Something is true near c means that it is true on some open interval containing c. 4

Example 1 5

Example 1 5

Example 2 Figure 2 shows the graph of a function f with several extrema:

Example 2 Figure 2 shows the graph of a function f with several extrema: At a : At b : At c : At d : At e : absolute minimum is f(a) local maximum is f(b) local minimum is f(c) absolute (also local) maximum is f(d) 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 6

Practice! Define all local and absolute extrema of the graph below 7

Practice! Define all local and absolute extrema of the graph below 7

Critical Point 8

Critical Point 8

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

Counter examples: f’ exists but there is no local min or max f’ doesn’t exist but there is a local min or max 9

Extreme Value Theorem The following theorem gives conditions under which a function is guaranteed

Extreme Value Theorem The following theorem gives conditions under which a function is guaranteed to have extreme values. 10

Extreme Value Theorem: Absolute min and max The Extreme Value Theorem is illustrated below:

Extreme Value Theorem: Absolute min and max The Extreme Value Theorem is illustrated below: Figure 7 Note that an extreme value can be taken on more than once. 11

Extreme Value Theorem : Absolute min and max The Extreme Value Theorem says that

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 12

Finding Absolute Maximum and Minimum of f: To find an absolute maximum or minimum

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. 13

Practice Find the absolute values of the function and where they occur (worksheet 3.

Practice Find the absolute values of the function and where they occur (worksheet 3. 1, # 5) 14

3. 2 The Mean Value Theorem 15

3. 2 The Mean Value Theorem 15

Rolle’s Theorem 16

Rolle’s Theorem 16

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

Examples: Figure 1 shows the graphs of four such functions. (a) (b) (c) (d) Figure 1 17

The Mean Value Theorem This theorem is an extension of Rolle’s Theorem 18

The Mean Value Theorem This theorem is an extension of Rolle’s Theorem 18

Example Show that f(x) satisfies the Mean Value Theorem on [a, b] f (x)

Example Show that f(x) satisfies the Mean Value Theorem on [a, b] f (x) = x 3 – 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) 19

Example - proof f (2) = 6, f (0) = 0, and f (x)

Example - proof f (2) = 6, f (0) = 0, and f (x) = 3 x 2 – 1, so this equation becomes: 6 = (3 c 2 – 1)2 = 6 c 2 – 2 which gives that is, c = But c must lie in (0, 2), so 20

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

The Mean Value Theorem can be used to establish some of the basic facts of differential calculus. 21