Instructions: If you are asked to find a function, enter a function. If you are asked to find $x$- or $y$-values, enter either a number, a list of numbers separated by commas, or None if there aren't any solutions. Use interval notation if you are asked to find an interval or union of intervals, and enter { } if the interval is empty.

(a) Calculate the first derivative of $f$. Find the critical numbers of $f$, where it is increasing and decreasing, and its local extrema.
$f'(x) =$
Critical numbers $x =$
Increasing on the interval
Decreasing on the interval
Local maxima $x =$
Local minima $x =$

(b) Calculate the second derivative of $f$. Find where $f$ is concave up, concave down, and has inflection points.
$f''(x) =$
Concave up on the interval
Concave down on the interval
Inflection points $x =$

(c) Find any horizontal and vertical asymptotes of $f$.
Horizontal asymptotes $y =$
Vertical asymptotes $x =$

(d) The function $f$ is because for all $x$ in the domain of $f$, and therefore its graph is symmetric about the

(e) Sketch a graph of the function $f$ without having a graphing calculator do it for you. Plot the $y$-intercept and the $x$-intercepts, if they are known. Draw dashed lines for horizontal and vertical asymptotes. Plot the points where $f$ has local maxima, local minima, and inflection points. Use what you know from parts (a) and (b) to sketch the remaining parts of the graph of $f$. Use any symmetry from part (d) to your advantage. Sketching graphs is an important skill that takes practice, and you may be asked to do it on quizzes or exams.

You can earn partial credit on this problem.