Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = x^{3} - x^{2} - 5 x + 1$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 3 x^{2} - 2 x - 5$ . Solving $LaTeX: \displaystyle 3 x^{2} - 2 x - 5 = 0$ gives $LaTeX: \displaystyle x = \left[ -1, \ \frac{5}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( -1 \right) = -8$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(-1\right) = 4$ is a local maximum.
$LaTeX: \displaystyle f''\left( \frac{5}{3} \right) = 8$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(\frac{5}{3}\right) = - \frac{148}{27}$ is a local minimum.