Please answer the following questions about the function
[math]
Instructions: If you are asked to find a function,
enter a function. If you are asked to find [math] - or [math] -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 [math] . Find the critical
numbers of [math] , where it is increasing and decreasing, and its local
extrema.
[math]
Critical numbers [math]
Increasing on the interval
Decreasing on the interval
Local maxima [math]
Local minima [math]
(b) Calculate the second derivative of [math] . Find where [math] is
concave up, concave down, and has inflection points.
[math]
Concave up on the interval
Concave down on the interval
Inflection points [math]
(c) Find any horizontal and vertical asymptotes of [math] .
Horizontal asymptotes [math]
Vertical asymptotes [math]
(d) The function [math] is
?
even
odd
neither
because
?
f(-x) = f(x)
f(-x) = - f(x)
not applicable
for all [math] in the domain of [math] , and therefore its graph is symmetric about the
?
x-axis
y-axis
origin
line y=x
not applicable
(e) Sketch a graph of the function [math] without having a graphing calculator do it for you. Plot the [math] -intercept and the [math] -intercepts, if they are known. Draw dashed lines for horizontal and vertical asymptotes. Plot the points where [math] 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 [math] . 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.