We are given the functions $$ f(x)=\frac{12x^2-16}{x^3},\qquad f'(x)=-\frac{12(x^2-4)}{x^4},\qquad f''(x)=\frac{24(x^2-8)}{x^5}. $$ The questions below are about the function $f(x)$. Answer parts (1) through (10) below. If the requested feature is missing, then explain why. Be sure to include the work/test that you used to rigorously reach your conclusion. It is not sufficient to refer to the graph.

(1) State the function's domain.
x^3 doesnt not equal 0, so the domain is: (-infintiy, 0) Union (0, infinity)

(2) Find all $x$- and $y$-intercepts.
x:set numerator equal to zero 12x^2-16=0, x=(+-4/√3)
y: set deom equal to zero -16/0= Y intercepts dont exist, there is no limit

(3) Find all equations of horizontal asymptotes.
x²/x³=2/3 the degree of the numerator is < denom, so the horizontal asymtope is x=0

(4) Find all equations of vertical asymptotes.
x³=0 so the vertical asymtope is x=0

(5) Find the interval(s) where $f$ is increasing.
- + | + -
— — — — — — --
-2 0 2
Since the pattern is -,+,0,+,- the interval is (-2,0), (0,2)

(6) Find the $x$-value(s) of all local maxima. (Find exact values, and not decimal representations)
Using the little line graph above it is clear the max of this graph goes up to 2 only so x=2 is the local maxima.

(7) Find the $x$-value(s) of all local minima. (Find exact values, and not decimal representations)
Based off the little line graph above it only goes down to -2 at the lowest so x=-2

(8) Find the interval(s) on which the graph is concave downward.
F''(x)=24( x^2 -8)/ x^5
The graph concaves down at (-infinty, -2√2), (0, 2√2)

(9) State the $x$-value(s) of all inflection points. (Find exact values, and not decimal representations)
For inflection points they are +-2√2

(10) Include a sketch of the graph of $y=f(x)$. Plot the different segments of the graph using the color code below.

blue: $f'>0$ and $f''>0$
red: $f'<0$ and $f''>0$
black: $f'>0$ and $f''<0$
gold: $f'<0$ and $f''<0$

(In Desmos you could restrict the plot $y=f(x)$ on the interval $[2,3]$ by typing $y=f(x){2\le x\le 3}$.) Be sure to set the bounds on the graph so that the features of the graph that you listed above are easy to see. :::