Determine the intervals on which the given function is concave up or down and find the point of inflection

Question

Determine the intervals on which the given function is concave up or down and find the point of inflection.

Let f(x) = x(x−4√x)

The x-coordinate of the point of inflection is: ____

The interval on the left of the inflection point is: ____ , and on this interval f is: __ concave up? or down? __

The interval on the right is: ____ , and on this interval f is: __ concave up? or down? __

Answer 1

f(x) = x² - 4x^3/2; f'(x) = 2x - 6x^1/2; f''(x) = 2 - 3/x^1/2_;

Domain of f(x): x ≥ 0. Let f''(x) = 0; 2x^1/2 = 3; x = 9/4. f'(x) DNE for x = 0 and at x = 0 we have not inflection point, because because x < 0 not in domain.

for 0 <x< 9/4 f''(x) < 0 and graph concave down or (0, 9/4)

for x > 9/4 f''(x) > 0 and graph concave up.or (9/4, ∞)

f(9/4) = -135/16. So, point (9/4, -135/16) is point of inflection