Graphing Problem!

[codyski95]

f(x)=x^(1/3)*(x+90)^(2/3)
This problem is a graphing problem where I need to find where it is increasing, decreasing, concave up and down, and inflection points. I have the critical numbers, but when I type it in on webwork, it doesnt seem to like my answers. The critical numbers I got are -90, -30, and 0. Also needs to be in interval notation.

IF anyone could help that would be greatly appreciated!!!!

 

[Ishuda]

f(x)=x^(1/3)*(x+90)^(2/3)
This problem is a graphing problem where I need to find where it is increasing, decreasing, concave up and down, and inflection points. I have the critical numbers, but when I type it in on webwork, it doesnt seem to like my answers. The critical numbers I got are -90, -30, and 0. Also needs to be in interval notation.

IF anyone could help that would be greatly appreciated!!!!

Go to the definitions:
Increasing: f' > 0,
Decreasing: f' < 0,
concave up: f'' > 0
concave down: f'' < 0
inflection point: f'' = 0

f = \(\displaystyle x^{1/3} (x+90)^{2/3}\)
f' = \(\displaystyle \frac{1}{3} [ x^{-2/3} (x+90)^{2/3} + 2 x^{1/3} (x+90)^{-1/3}]\)
=\(\displaystyle \frac{1}{3} \frac{ x+90 + 2 x}{x^{2/3} (x+90)^{1/3}}\)
=\(\displaystyle \frac{ x+30}{x^{2/3} (x+90)^{1/3}}\)
So f'=0 at x=-30 is correct for one number

If x < -30, f' is negative; if x > -30, f' is positive.

If a < x < b then x belongs to (a,b). If a \(\displaystyle \le\) x < b then x belongs to [a,b).

Oh, and \(\displaystyle \infty\) is always open, i.e. has the ( or ).