# If ( x+2) / x, what are the points of inflection, concavity and critical points?

Apr 6, 2018

#### Explanation:

Let $f \left(x\right) = \frac{x + 2}{x}$

The domain of $f \left(x\right)$ is $x \in \mathbb{R} - \left\{0\right\}$

The critical points are when $f ' \left(x\right) = 0$

$f ' \left(x\right) = \frac{1 \cdot x - \left(x + 2\right) + 1}{{x}^{2}} = - \frac{2}{x} ^ 2$

$f ' \left(x\right) \ne 0$

There are no critical points

The points of inflection are when $f ' ' \left(x\right) = 0$

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

$f ' ' \left(x\right) \ne 0$

There are no points of inflections

The concavity will depend on the sign of $f ' ' \left(x\right)$

Let's build a sign chart

$\textcolor{w h i t e}{a a a a}$$\text{Interval}$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , 0\right)$$\textcolor{w h i t e}{a a a a}$$\left(0 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$\text{Sign " f''(x)}$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$\text{Concavity}$$\textcolor{w h i t e}{a a a}$$c o n c a v e$$\textcolor{w h i t e}{a a a a}$$c o n v e x$

graph{(x+2)/x [-10, 10, -5, 5]}