Maxima & minima

Where a function peaks or bottoms out — set the derivative to zero.

Maxima and minima sit where the derivative is zero (or undefined). Critical points.

Drag the point along the curve

a = 1b = 0c = -3

y = a·x·x·x + b·x·x + c·xat x = 1: slope ≈ 0

Second derivative test

f' (c) = 0 an df ″ (c) > 0 \Rightarrow l oc a l min; f' (c) = 0 an df ″ (c) < 0 \Rightarrow l oc a l max

If f″(c) = 0 the test is inconclusive — fall back to a sign chart of f′.

Your turn

Find and classify the critical points of f(x) = x³ − 3x.

On a closed interval, also check the endpoints — the absolute max or min can sit at an endpoint even where f′ ≠ 0.

Steps