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

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = 21 x^{2} + 10 x + 1$ . Solving $LaTeX: \displaystyle 21 x^{2} + 10 x + 1 = 0$ gives $LaTeX: \displaystyle x = \left[ - \frac{1}{3}, \ - \frac{1}{7}\right]$ . 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 $LaTeX: \displaystyle f\left(- \frac{1}{3}\right) = \frac{80}{27}$ is a local maximum.
$LaTeX: \displaystyle f''\left( - \frac{1}{7} \right) = 4$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(- \frac{1}{7}\right) = \frac{144}{49}$ is a local minimum.