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