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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 3 x^{2} - 2 x - 1$ . Solving $LaTeX: \displaystyle 3 x^{2} - 2 x - 1 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ 1\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{1}{3} \right) = -4$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = - \frac{76}{27}$ is a local maximum.
$LaTeX: \displaystyle f''\left( 1 \right) = 4$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(1\right) = -4$ is a local minimum.