Mathematics

Mathematics

Problem

The derivative of a differentiable function is f′(x)=3x2−kf'(x)=3x^{2}-k, where k>0k>0. State how many critical points ff has, identify their xx–coordinates, and indicate whether each is a relative maximum or relative minimum. Justify your identification using the sign of f′(x)f'(x) on the intervals determined by the critical points.
Intuitive explanation

Critical points occur where the derivative equals zero, so you're solving 3x2−k=03x^2 - k = 0 for the values where the function's slope flips from positive to negative (or vice versa). Once you find those xx-values, use a sign chart: test f′(x)f'(x) on each interval created by the critical points to see whether the function is increasing or decreasing nearby. If f′f' changes from positive to negative, you have a local max; if it flips from negative to positive, you have a local min.

Intuitive explanation

Critical points occur where the derivative equals zero, so you're solving 3x2−k=03x^2 - k = 0 for the values where the function's slope flips from positive to negative (or vice versa). Once you find those xx-values, use a sign chart: test f′(x)f'(x) on each interval created by the critical points to see whether the function is increasing or decreasing nearby. If f′f' changes from positive to negative, you have a local max; if it flips from negative to positive, you have a local min.

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