# For what values of x is f(x)=-2x- (x+3)^ (-2/3) concave or convex?

Jun 18, 2017

$x \in \left(- \infty , - 3\right)$: $f \left(x\right)$ is concave down

$x \in \left(- 3 , + \infty\right)$: $f \left(x\right)$ is concave down

#### Explanation:

Find the first derivative, which is the slope of the curve

$f ' \left(x\right) = - 2 + \frac{2}{3} {\left(x + 3\right)}^{- \frac{5}{3}}$

The second derivative, is the rate of change of the curve

$f ' ' \left(x\right) = - \frac{10}{9} {\left(x + 3\right)}^{- \frac{8}{3}} = \frac{- 10}{9 \sqrt[3]{{\left(x + 3\right)}^{8}}}$

Whenever $f ' ' \left(x\right) < 0$, the curve is concave downward (like a frown face) and whenever $f ' ' \left(x\right) > 0$, the curve is concave upward (like a smiley face).

Testing points around $x = - 3$ (where $f ' ' \left(x\right)$ divides by zero).

$x < - 3$: All points are negative; Concave down

$x > - 3$: All points are negative; Concave down