# How do you determine the intervals for which the function is increasing or decreasing given f(x)=1/(x+1)-4?

Sep 8, 2017

The intervals of decreasing are $\left(- \infty , - 1\right) \cup \left(- 1 , + \infty\right)$

#### Explanation:

We need

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

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

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R} - \left\{- 1\right\}$

Therefore, the derivative of $f \left(x\right)$ is

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

$\forall x \in {D}_{f} \left(x\right) , f ' \left(x\right) < 0$

There are no critical values, and the variation table is

$\textcolor{w h i t e}{a a a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a}$$\left(- \infty , - 1\right)$$\textcolor{w h i t e}{a a a a}$$\left(- 1 , + \infty\right)$

$\textcolor{w h i t e}{a a a a}$$s i g n f ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$↘$\textcolor{w h i t e}{a a a a a a a a a a a}$↘

You can also, calculate the second derivative and determine the concavity.

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