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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 3 x^{2} + 8 x + 3$ . Solving $LaTeX: \displaystyle - 3 x^{2} + 8 x + 3 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ 3\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = 10$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{95}{27}$ is a local minimum.
$LaTeX: \displaystyle f''\left( 3 \right) = -10$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(3\right) = 15$ is a local maximum.