How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for f(x)=2x^(5/3)-5x^(4/3)?

1 Answer
Nov 14, 2016

Please see below.

Explanation:

f'(x) = 10/3x^(2/3)-20/3x^(1/3)

f''(x) = 20/9x^(-1/3)-20/9x^(-2/3))

= 20/9x^(-2/3)(x^(1/3)-1)

= 20/9 * ((root(3)x-1)/(root(3)x^2))

Sign of f''

The denominator is always positive and the numerator (hence the second derivative) is negative for x < 1 and positive for x > 1.

There is a point of inflection at x=1

f is defined and continuous on (oo,oo).

From the sign of f'' we see that f is concave down (concave) on (-oo,1) and

f is concave up (convex) on (1,oo).