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. 