Local Extreme Points

Objectives

Students will be able to• Find relative maximum and minimum

points of a function.

First-Derivative Test for Local Extrema

Suppose c is a critical point for y = f(x)

• If f’ (x) > 0 throughout some interval (a, c) to the left of c and f’ (x) < 0 throughout some interval (c, b) to the right of c, then x = c is a local maximum point for the function f.

AND

First-Derivative Test for Local Extrema

Suppose c is a critical point for y = f(x)

• If f’ (x) < 0 throughout some interval (a, c) to the left of c and f’ (x) > 0 throughout some interval (c, b) to the right of c, then x = c is a local minimum point for the function f.

AND

First-Derivative Test for Local Extrema

Suppose c is a critical point for y = f(x)

• If f’ (x) > 0 (or f’ (x) < 0) throughout some interval (a, c) to the left of c and throughout some interval (c, b) to the right of c, then x = c is not a local minimum point for the function f.

Second Derivative Test

Let f be a twice differentiable function in an interval I, and let c be an interior point of I. Then•if f’ (c) = 0 and f’’ (c) < 0, then x = c is a strict local maximum point.•if f’ (c) = 0 and f’’ (c) > 0, then x = c is a strict local minimum point.•if f’ (c) = 0 and f’’ (c) = 0, then no conclusion can be drawn.

Example 1

Find the locations and values of all local extrema for the function with the graph

Example 2

Find the locations and values of all local extrema for the function with the graph

Example 3Suppose that the graph to the right is the graph of f’ (x) , the derivative of f(x). Find the locations of all relative extrema and tell whether each extremum is a relative maximum or minimum

Example 4

Find the critical points for the function below and determine if they are relative maximum or minimum points or neither.

31292)( 23 xxxxf

Example 5

Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

32

376)( xxf

Example 6

Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

3)( 8 xexxf

Example 7For the cost function

and the price function

find

qqC 1480)(

qp 258

a. the number, q, of units that produces a maximum profit.

b. the price, p, per unit that produces maximum profit.

c. the maximum profit, P.

Example 8Suppose that the cost function for a product is given by

find the production level (i.e. value of x) that will produce the minimum average cost per unit .

78138002.0)( 3 xxxC

)(xC

In Summary

To find local extrema, we need to look at the following types of points:

i. Interior point in an interval I where f’ (x) = 0

ii.End points of I (if included in I) iii.Interior points in I where f’ (x)

does not exist