# How do you find points of inflection and determine the intervals of concavity given y=2x^(1/3)+3?

Jul 23, 2017

Investigate the sign of $y ' '$

#### Explanation:

$y ' = \frac{2}{3} {x}^{- \frac{2}{3}}$

$y ' ' = - \frac{4}{9} {x}^{- \frac{5}{3}} = \frac{- 4}{9 {x}^{\frac{5}{3}}}$

Note that the sign of ${x}^{\frac{5}{3}}$ is the same as that of $x$, so

$y ' '$ is positive left of $x = 0$ and the graph is concave up (convex)

and $y ' '$ is negative right of $0$ and the graph is concave down (concave).

The concavity changes at $x = 0$ whic is in the domain of the function, so there is a inflection point at $x = 0$ which make $y = 3$

$\left(0 , 3\right)$ is the only inflection point.