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f(x) = x³ − 6x² + 9x + 1

local max · x=1local min · x=3

Mathematics S7 · Calculus

Find the stationary points of $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ .

First, differentiate: $f^{'} (x) = 3 x^{2} - 12 x + 9$ .

Set $f^{'} (x) = 0$ : $3 x^{2} - 12 x + 9 = 0$ , which simplifies to $x^{2} - 4 x + 3 = 0$ , so $(x - 1) (x - 3) = 0$ .

Stationary points at $x = 1$ and $x = 3$ . Using $f^{''} (x) = 6 x - 12$ : $f^{''} (1) = - 6 < 0$ (local max), $f^{''} (3) = 6 > 0$ (local min).

The graph below confirms the local max at x=1 and local min at x=3.

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