# Is f(x) = (x+3)^(2/3) - 6 concave or convex?

Oct 10, 2016

$f \left(x\right)$ has a convex hypograph.

#### Explanation:

For $x \in \left(- 3 , \infty\right)$ as we can see, $f \left(x\right) = {\left(x + 3\right)}^{\frac{2}{3}} - 6$ has a convex hypograph. This can be stated observing that in the same range, $f ' \left(x\right) = \frac{2}{3 {\left(3 + x\right)}^{\frac{1}{3}}}$ has a positive strictly decreasing value. Further rigorous proof can be established by stating that if
$\left({x}_{1} , f \left({x}_{1}\right)\right)$ and $\left({x}_{2} , f \left({x}_{2}\right)\right)$ are two hypograph points, then

$\left({x}_{1} + \lambda \left({x}_{2} - {x}_{1}\right) , f \left({x}_{1} + \lambda \left({x}_{2} - {x}_{1}\right)\right)\right)$ for $\lambda \in \left[0 , 1\right]$ is also a hypograph point.

Attached the hypograph set