# 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)?

Nov 14, 2016

#### Explanation:

$f ' \left(x\right) = \frac{10}{3} {x}^{\frac{2}{3}} - \frac{20}{3} {x}^{\frac{1}{3}}$

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

$= \frac{20}{9} {x}^{- \frac{2}{3}} \left({x}^{\frac{1}{3}} - 1\right)$

$= \frac{20}{9} \cdot \left(\frac{\sqrt[3]{x} - 1}{{\sqrt[3]{x}}^{2}}\right)$

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 $\left(\infty , \infty\right)$.

From the sign of $f ' '$ we see that $f$ is concave down (concave) on $\left(- \infty , 1\right)$ and

$f$ is concave up (convex) on $\left(1 , \infty\right)$.