A continuous function on a closed interval attains its maximum, but f'(x) = 0 need not hold there. Why can the maximum sit somewhere the derivative is not zero?

Question

Accepted Answer

At an endpoint, where the function can still be climbing

Answer

At a discontinuity, which is why continuity is required

Answer

Nowhere else: a maximum always has zero derivative

Answer

At a point of inflection, where the second derivative vanishes

A continuous function on a closed interval attains its maximum, but $f'(x) = 0$ need not hold there. Why can the maximum sit somewhere the derivative is not zero?

Practise Extrema and optimization

More questions on Extrema and optimization

A continuous function on a closed interval attains its maximum, but f′(x)=0f'(x) = 0f′(x)=0 need not hold there. Why can the maximum sit somewhere the derivative is not zero?

Practise Extrema and optimization

More questions on Extrema and optimization

A continuous function on a closed interval attains its maximum, but $f'(x) = 0$ need not hold there. Why can the maximum sit somewhere the derivative is not zero?