# What are the local extrema, if any, of f(x)= 2x+15x^(2/15)?

Jul 23, 2017

Local maximum of 13 at 1 and local minimum of 0 at 0.

#### Explanation:

Domain of $f$ is $\mathbb{R}$

$f ' \left(x\right) = 2 + 2 {x}^{- \frac{13}{15}} = \frac{2 {x}^{\frac{13}{15}} + 2}{x} ^ \left(\frac{13}{15}\right)$

$f ' \left(x\right) = 0$ at $x = - 1$ and $f ' \left(x\right)$ does not exist at $x = 0$.

Both $- 1$ and $9$ are in the domain of $f$, so they are both critical numbers.

First Derivative Test:

On $\left(- \infty , - 1\right)$, $f ' \left(x\right) > 0$ (for example at $x = - {2}^{15}$)
On $\left(- 1 , 0\right)$, $f ' \left(x\right) < 0$ (for example at $x = - \frac{1}{2} ^ 15$)

Therefore $f \left(- 1\right) = 13$ is a local maximum.

On $\left(0 , \infty\right)$, $f ' \left(x\right) > 0$ (use any large positive $x$)

So $f \left(0\right) = 0$ is a local minimum.