# What are the local extrema, if any, of f(x)= –2x^3 + 6x^2 + 18x –18?

Mar 22, 2016

Maximum f is $f \left(\frac{5}{2}\right)$ = 69.25. Minimum f is $f \left(- \frac{3}{2}\right)$ = 11.25.
$\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right) = - 6 {x}^{2} + 12 x + 18 = 0$, when $x = \frac{5}{2} \mathmr{and} - \frac{3}{2}$
The second derivative is $- 12 x + 12 = 12 \left(1 - x\right) < 0$ at $x = \frac{5}{2}$ and > 0 at x = $\frac{3}{2}$.
So, f($\frac{5}{2}$) is the local (for finite x) maximum and f($- \frac{3}{2}$) is the local (for finite x) minimum.
As $x \to \infty , f \to - \infty$ and as $x \to - \infty , f \to + \infty$..