Optimization Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

Find the critical points of

f(x) = 2x^3 - 9x^2 + 12x - 4

and classify them.

1
$f'(x) = 6x^2 - 18x + 12 = 6(x^2 - 3x + 2) = 6(x-1)(x-2)$ .
2
Critical points: $x = 1$ and $x = 2$ .
3
$f''(x) = 12x - 18$ . At $x=1$ : $f''(1) = -6 < 0$ (local max). At $x=2$ : $f''(2) = 6 > 0$ (local min).

Answer

Local maximum at

x = 1

; local minimum at

x = 2

Factor the first derivative to find critical points, then use the second derivative test to classify each one as a local max or min.

About Optimization

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

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