Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 3x(1/3) + 6x(4/3). You must justify your answer using an analysis of f '(x) and f "(x)

Question

I got -1/8 for a minimum point and 1/4 for inflection is this right? Do I need a y value?

Answer 1

I assume you mean

y = 3x^(1/3) + 6x^(4/3) = 3(2x+1) x^(1/3)
y' = x^(-2/3) + 8x^(1/3) = (8x+1) x^(-2/3)
y" = -2/3 x^(-5/3) + 8/3 x^(-2/3) = 2/3 (4x-1) x^(-5/3)

y'=0 at x = -1/8
y"(-1/8) < 0 so y(-1/8) is a maximum

y"=0 at x = 1/4
so that is an inflection point.

The question does not ask for y values.