# How do you sketch the curve y=(x+5)^(1/4) by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Jun 28, 2017

The function $f \left(x\right) = {\left(x + 5\right)}^{\frac{1}{4}}$ is defined and continuous for $\left(x + 5\right) > 0$, that is for $x \in \left[- 5 , + \infty\right]$.

At the boundaries of the domain we have:

${\lim}_{x \to {5}^{+}} {\left(x + 5\right)}^{\frac{1}{4}} = 0$

${\lim}_{x \to + \infty} {\left(x + 5\right)}^{\frac{1}{4}} = \infty$

inside the domain the function has only positive values.

Evaluate the derivatives:

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

$f ' ' \left(x\right) = - \frac{3}{8} {\left(x + 5\right)}^{- \frac{7}{4}}$

So we have that:

$f ' \left(x\right) > 0$ for $x \in \left(- 5 , + \infty\right)$ and ${\lim}_{x \to {5}^{+}} f ' \left(x\right) = \infty$

$f ' ' \left(x\right) < 0$ for $x \in \left(- 5 , + \infty\right)$

We can then conclude that $f \left(x\right)$ is strictly increasing and has no relative maxima or minima and is concave down in its domain:

graph{(x+5)^(1/4) [-10, 10, -5, 5]}