Find the local maximum and minimum of $LaTeX: \displaystyle f(x) = - 5 x^{3} + 19 x^{2} - 7 x + 3$ .

To find the critical numbers solve $LaTeX: \displaystyle f'(x) = 0$ . The derivative is $LaTeX: \displaystyle f'(x) = - 15 x^{2} + 38 x - 7$ . Solving $LaTeX: \displaystyle - 15 x^{2} + 38 x - 7 = 0$ gives $LaTeX: \displaystyle x = \left[ \frac{1}{5}, \ \frac{7}{3}\right]$ . Using the 2nd derivative test gives:
$LaTeX: \displaystyle f''\left( \frac{1}{5} \right) = 32$ which is greater than zero, so the function is concave up and $LaTeX: \displaystyle f\left(\frac{1}{5}\right) = \frac{58}{25}$ is a local minimum.
$LaTeX: \displaystyle f''\left( \frac{7}{3} \right) = -32$ which is less than zero, so the function is concave down and $LaTeX: \displaystyle f\left(\frac{7}{3}\right) = \frac{718}{27}$ is a local maximum.