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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 1 - 9 x^{2}$ . Solving $LaTeX: \displaystyle 1 - 9 x^{2} = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ \frac{1}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 6$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = \frac{16}{9}$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{1}{3} \right) = -6$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{1}{3}\right) = \frac{20}{9}$ is a local maximum.