Calculus problem

Question

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).

Answer 1

f(x) = 3x^(1/3) + 6x^(4/3)

First, find the roots. Set = to 0, and divide both sides by 3x^1/3 :

1 = -2x, x = -1/2. So the roots are 0 and -1/2. Note that, for large + or - x, the curve heads to + infinity.

To find the critical points, start by taking the 1st derivative and setting equal to 0:

f'(x) = x^-2/3 + 8x^1/3 = 0

divide both sides by x^1/3 :

x^-1 + 8 = 0

So the critical points are x = 0 and x = -1/8

Since the basic function heads to infinity for large +/- x, we can guess that x = -1/8 is a minimum.

To determine what happens at x = 0, take the 2nd derivative and set to 0:

f"(x) = -2/3 *x^-5/3 + 8/3 * x^-2/3 = 0

divide both sides by x^-2/3 / 2/3:

-x^-1 + 4 = 0

So, there are critical points at x = 0, and x = 1/4

To see what happens at all these points, plot the original function