# How do you find all local maximum and minimum points using the second derivative test given y=(x+5)^(1/4)?

Feb 2, 2017

${\left(x + 5\right)}^{\frac{1}{4}}$ is a monotonically increasing function and there are no maxima or minima.

#### Explanation:

We observe that the domain of $x$ is $\left[- 5 , \infty\right)$

As $y = {\left(x + 5\right)}^{\frac{1}{4}}$

$y ' = \frac{1}{4} {\left(x + 5\right)}^{\left(1 - \frac{1}{4}\right)} = \frac{1}{4} {\left(x + 5\right)}^{- \frac{3}{4}} = \frac{1}{4 {\left(x + 5\right)}^{\frac{3}{4}}}$

It is apparent that $x + 5$ is always positive within the domain $\left[- 5 , \infty\right)$

Further y''=1/4xx-3/4xx(x+5)^(-7/4)=(-3)/(16(x+5)^(7/4) and is always negative and hence

${\left(x + 5\right)}^{\frac{1}{4}}$ is a monotonically increasing function and there are no maxima or minima.
graph{(x+5)^(1/4) [-8.71, 11.29, -3.64, 6.36]}