Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 9 x^{3} - 18 x^{2} - 9 x - 2$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 27 x^{2} - 36 x - 9$ . Solving $LaTeX: \displaystyle - 27 x^{2} - 36 x - 9 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 18$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(-1\right) = -2$ is a local minimum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = -18$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{2}{3}$ is a local maximum.