   Chapter 3.1, Problem 71E

Chapter
Section
Textbook Problem

# Prove Fermat’s Theorem for the case in which f has a local minimum at c.

To determine

To prove:

Fermat’s theorem for the case in which  f has a local minimum at c

Explanation

1) Concept:

Use the concept of local maximum and minimum and  Fermat’s theoremfor local maximum 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) Fermat’s theorem:

If  f has a local maximum or minimum at c and if f'(c) exists, then f'c=0.

4) Given:

f has a local minimum value at c

5) Calculation:

It is given 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 f(c)f(x) by -1

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