Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 5 x^{3} - 10 x^{2} - 5 x - 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 15 x^{2} - 20 x - 5$ . Solving $LaTeX: \displaystyle - 15 x^{2} - 20 x - 5 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ - \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = 10$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(-1\right) = -1$ is a local minimum.
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = -10$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{7}{27}$ is a local maximum.