Quick Math Calculus Help Please!?

Question

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 9x^(1/3) + (9/2)x^(4/3). You must justify your answer using an analysis of f ′(x) and f ′′(x).

Answer 1

f(x) = 9x^(1/3) + (9/2)x^(4/3)

 

You get extrema when f'(x) goes to zero, and inflection points when f"(x) goes to zero.

 

f'(x) = 3x^(-2/3) + 6x^(1/3) = 3x^(-2/3)(1 + 2x)

 

That goes to zero when x = -1/2.

 

f"(x) = -2x^(-5/3)(1 + 2x) + 6x^(-2/3)

= -2x^(-5/3)(1 + 5x)

 

And that goes to zero when x = -1/5.

 

Sanity Check: I ran this problem through Wolfram Alpha. I got the same real values for x when f'(x)=0 and f"(x)=0.