Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 3 x^{3} - 9 x^{2} - 9 x - 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 9 x^{2} - 18 x - 9$ . Solving $LaTeX: \displaystyle - 9 x^{2} - 18 x - 9 = 0$ gives $LaTeX: \displaystyle x = \left[ -1\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 0$ The 2nd derivative test fails.