Question

Consider the function

f(x)=

x³

x² − 4

Express the domain of the function in interval notation:

Find the y-intercept: y=

.
Find all the x-intercepts (enter your answer as a comma-separated list): x=

.
On which intervals is the function positive?

On which intervals is the function negative?

Does f have any symmetries?

f is even;f is odd;    f is periodic;None of the above.

Find all the asymptotes of f (enter your answers as equations):

Vertical asymptote (left):

;
Vertical asymptote (right):

;
Asymptote at

x → ∞

:

.

Determine the derivative of f.

f'(x)=

On which intervals is f increasing/decreasing? (Use the union symbol and not a comma to separate different intervals; if the function is nowhere increasing or nowhere decreasing, use DNE as appropriate).

f is increasing on

.
f is decreasing on

.

List all the local maxima and minima of f. Enter each maximum or minimum as the coordinates of the point on the graph. For example, if f has a maximum at

x=3 and f(3)=9, enter (3,9)

in the box for maxima. If there are multiple maxima or minima, enter them as a comma-separated list of points, e.g.

(3,9),(0,0),(4,7)

. If there are none, enter DNE.

Local maxima:

.
Local minima:

.

Determine the second derivative of f.

f''(x)=

On which intervals does f have concavity upwards/downwards? (Use the union symbol and not a comma to separate different intervals; if the function does not have concavity upwards or downwards on any interval, use DNE as appropriate).

f is concave upwards on

.
f is concave downwards on

.

List all the inflection points of f. Enter each inflection point as the coordinates of the point on the graph. For example, if f has an inflection point at

x=7 and f(7)=−2, enter (7,−2)

in the box. If there are multiple inflection points, enter them as a comma-separated list, e.g.

(7,−2),(0,0),(4,7)

. If there are none, enter DNE.

Does the function have any of the following features? Select all that apply.

Removable discontinuities (i.e. points where the limit exists, but it is different than the value of the function)Corners (i.e. points where the left and right derivatives are defined but are different)Jump discontinuities (i.e. points where the left and right limits exist but are different)Points with a vertical tangent line

Draw the graph by hand and with all the details. You should clearly indicate all the relevant features of the function, including information that may not have been requested here explicitly, for example the limits at the edges of the domain and the slopes of tangent lines at interesting points (e.g. inflection points).

Answer 1