# Thread: Function: Find Increasing/Decreasing, Concavity up down. etc.

1. ## Function: Find Increasing/Decreasing, Concavity up down. etc.

f(x)=x^3-8x+16/x

For the following function, find the

Domain: all reals but 0
Symmetry: Origin Symmetry
intercepts - y - none
- x = +-2
relative extrema :
Taking f'(x) and solving for 0 to get the critical numbers
f'(x) = 3x^2 - 16/x^2 - 8
setting = to 0, gives critical numbers of +-2
Plugging in critical numbers into f(x)
f(-2) = 0
f(2) = 0
can't tell which is max or min, so make a number line, test points to left of -2, in between -2 and 2, and to the right of 2
f(-3) = -8.333
f(-2) = 0
f(-1) = -9
f(1) = 9
f(2)=0
f(3) = 8.333

This means that a maximum is at -2,0 and minimum at 2,0

Is that correct for the relative extrema?
intervals of increasing/decreasing behavior
For the intervals, once again, i found the critical number, made the number ilne and picked test points
From (-∞ to -2) the graph was increasing, (-2 to 0) decreasing , (0 to 2) decreasing, and (2 to ∞) increasing

whats the right way to do this problem
inflection points
i know we need to take the second derivative and set = to 0 but it is not working out? Are there no inflection points? and if there are no inflection points, how can there be any intervals of concavity? Can there?

f"(x) = 6x+32/x^3
cant solve for f"(x) = 0

What do i do?
intervals of concave up/down

2. Originally Posted by NandP
f(x)=x^3-8x+16/x
Hi NandP,

Is x=0 part of the function domain?

What is f(0) ?

What is the shape of the graph now, given that your work is correct.
You can now deduce your concave regions.

3. f(0) is undefined. What does that mean about the inflection points though? I dont think there are any inflection points.. and as for concavity, i think the graph just concaves down from -∞ to 0 and concaves up from 0 to ∞..if you graph it, its just two parabolas, one opening down from -∞ to 0 and one opening up from 0 to ∞,

i sitll dont know what to do for the last two, inflection and intervals

4. Sorry, I don't normally use terms like "undefined",
since it doesn't really say much.

For your function, x=0 causes 16/x to shoot to infinity,
plus infinity for x>0 and minus infinity for x<0.
f(x) approaches infinity because the 16/x term is dominant for x near zero.

Your graph 'looks like' a pair of parabolas.