Mathematics

Problem

A function

f

is differentiable on

[a,b]

. Explain briefly why a point

c

inside

(a,b)

where

f'(x)>0

for values of

x

both just less than and just greater than

c

cannot be the location of an absolute maximum of

f

[a,b]

Intuitive explanation

The key insight is that if $f'(x) > 0$ on both sides of $c$ , the function is strictly increasing through that point—it's going upward as you pass through $c$ . This means $f(c)$ is smaller than values of $f$ slightly to the right of $c$ , so $c$ cannot be where $f$ reaches its largest value on the interval. An absolute maximum must occur where the function stops increasing, either at an endpoint or where the derivative changes sign.

Intuitive explanation

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