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Chapter 3: Q. 91 (page 263)

Prove part (b) of Theorem 3.8: With hypotheses as stated in the theorem, if x=c is a critical point of f, where f'(x)<0 to the left of c and f'(x)>0 to the right of c, then f has a local minimum at x=c.

Short Answer

Expert verified

The part(b) of theorem3.8 is proved.

Step by step solution

01

Step 1. Given Information 

We are given a function f and theorem 3.8.

02

Step 2. Proving the statement 

Suppose f'(x)<0for x鈭�(a,c)and f'(x)>0for x鈭�(c,b), that is suppose f is decreasing on (a,c]and increasing on [c,b). We will show that f(c)鈮�f(x)for all x鈭�(a,b) which will tell us that f has local minimum at x=c.

Given that x鈭�(a,b), there will be 3cases to consider.

First if x=cthen clearly f(c)=f(x).

Second if a<x<c then since f is decreasing on (a,c]we have f(x)>f(c).

Third if c<x<b then since f is increasing on [c,b), we have f(x)>f(c).

In all the three cases we have f(x)鈮�f(c) and therefore f has a local minimum at x=c.

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