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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 15 x^{2} - 10 x - 5$ . Solving $LaTeX: \displaystyle 15 x^{2} - 10 x - 5 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ 1\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = -20$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = \frac{79}{27}$ is a local maximum.
$LaTeX: \displaystyle f''\left( 1 \right) = 20$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(1\right) = -3$ is a local minimum.