   Chapter 3.1, Problem 70E

Chapter
Section
Textbook Problem

# If f has a local minimum value at c, show that the function g ( x )   = − f ( x ) has a local maximum value at c.

To determine

To show:

The function gx=-fx has a local maximum value at c if fhas a local minimum value at c.

Explanation

1) Concept:

Use the concept of local maximum and minimum to show the required result.

2) Definition:

The number f(c) is a local maximum value of f if  f(c)f(x) when x  is near c  and local minimum value of f if f(c)f(x) when x is near c.

3) Given:

f has a local minimum value at c.

4) Calculation:

Suppose that f has a local minimum value at c.

Therefore, by the concept of local minimum, f(c)f(x) when x  is near  c.

To show that the function gx=-fx has a local maximum value at  c, multiply both sides of the inequality f(c)f(x) by -1

So, -fc-f(

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