4.1 Extreme Values of Function

Calculus

Extreme Values of a function are created when the function changes from increasing to decreasing or from decreasing to increasing

Extreme value

decreasing

increasingincreasing

Extreme value

decreasing

dec

dec inc

Extreme value

Extreme value

inc dec

inc

dec

Extreme value

Extreme value

Extreme value

Absolute Minimum – the smallest function value in the domain

Absolute Maximum – the largest function value in the domain

Local Minimum – the smallest function value in an open interval in the domain

Local Maximum – the largest function value in an open interval in the domain

Classifications of Extreme Values

Absolute MinimumAbsolute Minimum

Absolute Maximum

Absolute Maximum

Local Minimum

Local Minimum

Local Minimum

Local Minimum

Local Minimum

Local Maximum

Local Maximum

Local Maximum

Local Maximum

Local Maximum

Absolute Minimum at c

c

Absolute Maximum at c

c

Definitions:

Local Minimum at c

ca b

Local Maximum at c

ca b

The number f(c) is called the maximum value of f on Df(c)

c d

The number f(d) is called the maximum value of f on D

f(d)

The maximum and minimum values of are called the extreme values of f.

Example1:

Example2:

Example3:

The Extreme Value Theorem (Max-Min Existence Theorem)

If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value.

a bc

Absolute maximum value: f(a)

Absolute minimum value: f(c)

The Extreme Value Theorem (Max-Min Existence Theorem)

If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value.

a bd

Absolute maximum value: f(c)

Absolute minimum value: f(d)

c

The Extreme Value Theorem (Max-Min Existence Theorem)

If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value.

Absolute maximum value: none

Absolute minimum value: f(d)

a bdc

F is not continuous at c.

Theorem does not apply.

The Extreme Value Theorem (Max-Min Existence Theorem)

If a function is continuous on a closed interval, [a, b], then the function will contain both an absolute maximum value and an absolute minimum value.

Absolute maximum value: f(c)

Absolute minimum value: f(d)

F is not continuous at c.

Theorem does not apply.

a bdc

Fermat’s Theorem is named after Pierre Fermat (1601–1665), a French lawyer who took up mathematics as a hobby. Despite his amateur status, Fermat was one of the two inventors of analytic geometry (Descartes was the other). His methods for finding tangents to curves and maximum and minimum values (before the inventionof limits and derivatives) made him a forerunner of Newton in the creation of differential calculus.

Sec 3.11 HYPERBOLIC FUNCTIONSThe following examples caution us against reading too much into Fermat’s Theorem. We can’t expect to locate extreme values simply by setting f’(x) = 0 and solving for x.

Exampe5: 3)( xxf Exampe6: xxf )(

WARNING Examples 5 and 6 show that we must be careful when using Fermat’s Theorem. Example 5 demonstrates that even when f’(c)=0 there need not be a maximum or minimum at . (In other words, the converse of Fermat’s Theorem is false in general.) Furthermore, there may be an extreme value even when f’(c)=0 does not exist (as in Example 6).

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.

A

F081

F083

F091

In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4):

F092

F091

F081

F081

F092

F081

F083

F083

a b c d

Which table best describes the graph?

Table A

Table B

Table C