Mathematics

Mathematics

Problem

A differentiable function ff satisfies f′(x)<0f'(x) < 0 for x<−2x < -2, f′(−2)=0f'(-2) = 0, f′(x)>0f'(x) > 0 for −2<x<1-2 < x < 1, f′(1)=0f'(1) = 0, and f′(x)<0f'(x) < 0 for x>1x > 1. Determine the number of relative (local) maxima of ff and state their xx-coordinate(s).
Intuitive explanation

The derivative tells you exactly where ff is increasing or decreasing, and the sign changes of f′f' reveal local extrema. When f′f' changes from negative to positive, you have a local minimum; when it flips from positive to negative, that's a local maximum. Trace through the signs: f′f' goes negative → zero → positive → zero → negative, so ff dips down, then climbs up, then falls again—giving you one local maximum where the climb stops.

Intuitive explanation

The derivative tells you exactly where ff is increasing or decreasing, and the sign changes of f′f' reveal local extrema. When f′f' changes from negative to positive, you have a local minimum; when it flips from positive to negative, that's a local maximum. Trace through the signs: f′f' goes negative → zero → positive → zero → negative, so ff dips down, then climbs up, then falls again—giving you one local maximum where the climb stops.

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