Practice question · Multiple choice
To find the maximum of a continuous function on a closed interval you must check the endpoints as well as the critical points. Why can the derivative alone miss the answer?
Hints
- Take f(x) = x on the interval from 0 to 1. Where is the maximum, and what is f' there?
- Ask what f'(c) = 0 actually detects.
Show the answer
B. Because a maximum at an endpoint need not have a horizontal tangent.
Why
The clearest case is f(x) = x on [0, 1]: the maximum is at the right endpoint and the derivative is 1 everywhere, so the critical-point search finds nothing. At an interior peak the function rises then falls, forcing a horizontal tangent between; at an endpoint there is no 'after'. It is also why the Extreme Value Theorem needs a closed interval.
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