Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = 5 x^{3} - x^{2} - x - 4$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 15 x^{2} - 2 x - 1$ . Solving $LaTeX: \displaystyle 15 x^{2} - 2 x - 1 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{5}, \ \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{5} \right) = -8$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{5}\right) = - \frac{97}{25}$ is a local maximum.
$LaTeX: \displaystyle f''\left( \frac{1}{3} \right) = 8$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(\frac{1}{3}\right) = - \frac{115}{27}$ is a local minimum.