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Chapter 3: Q.64 (page 249)

Prove the part of Theorem 3.3 that was not proved in the reading: If a function f has a local minimum at x=c, then either f(c)does not exist or f(c)=0.

Short Answer

Expert verified

The part that is not proved is proved by Theorem 3.3.

Step by step solution

01

Step 1. Given Information.

The function isf(x).

02

Step 2. Local Minimum.

Let x=cbe the location of the local minimum of f.

If f'(c)does not exist, then x=cis a critical point.

We assume f'(c)exists, then show that f'(c)=0.

As x=cis the location of local minimum of f, then there exists some 未>0such that,

f(c)鈮�f(x)for all x鈭�(c-未,c+未)

That is, f(x)-f(c)鈮�0.

If f'(c)exists such that f'(c)鈮�0and f'(c)鈮�0, then f'(c)must be equal to zero.

That is, f'(c)=0.

Hence, it is a functionfhas local minimum atx=c, then eitherf'(c)does not exist orf'(c)=0.

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