Question

Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x^(1/3) + 3x^(4/3). Justify your answer using an analysis of f ′(x) and f ′′(x).

Answer 1

The x-coordinate of the relative extrema of a function is determined by taking the first derivative of the function. In this case, f'(x) = 2 x^-2/3 + 4 x^1/3. We equate f'(x) to zero and the x-coordinates are The inflection point is determined by taking the second derivative of the original function. hence, x = -0.5f''(x) = -4/3 x ^ -5/3 + 4/3 x^-2/3 = 0; x of the inflection point is equal to 1.