Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 3 x^{3} - 3 x^{2} - x - 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 9 x^{2} - 6 x - 1$ . Solving $LaTeX: \displaystyle - 9 x^{2} - 6 x - 1 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 0$ The 2nd derivative test fails.