Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = x^{3} + 5 x^{2} + 7 x + 2$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 3 x^{2} + 10 x + 7$ . Solving $LaTeX: \displaystyle 3 x^{2} + 10 x + 7 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{7}{3}, \ -1\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( - \frac{7}{3} \right) = -4$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(- \frac{7}{3}\right) = \frac{5}{27}$ is a local maximum.
$LaTeX: \displaystyle f''\left( -1 \right) = 4$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(-1\right) = -1$ is a local minimum.