Subsections

• Background
• Exercises

First and second derivative tests

Background

Use the commands that you learned in the first four labs to complete these exercises.

Exercises

For each exercise below, do NOT look at a plot of the function. You may however, plot the derivative to help you if you need it.

1. For the function $$f(x) = \frac{x^2-x+9}{x^2-x-12}$$,
   1. Find all critical values (where the derivative is zero or undefined).
   2. Find intervals on which $f$ is increasing and the intervals on which it is decreasing.
2. For the function $$g(x)=\frac{1}{\sqrt{x^3+3x^2-9x+24}}$$
   1. Find all critical points. (You do not need to consider where the derivative is undefined here because there could never be a relative extrema at a point where $g'$ is undefined. Also remember a critical point implies finding corresponding $y$ values for each critical value.)
   2. Use the second derivative test to classify each point as a relative maximum or a relative minimum, again, without plotting the function.
3. Find the absolute extrema for the function $$h(x)=\sqrt{x}-x\sin(x)$$ on the closed interval $[0,7]$, again, without plotting the function.