# If  e^x / (5+e^x), what are the points of inflection, concavity and critical points?

Oct 27, 2017

The point of inflection is $\left(\ln 5 , \frac{1}{2}\right)$ and no critical points. The concavities are shown below

#### Explanation:

We need

$\left(\frac{u}{v}\right) ' = \frac{u ' v - u v '}{{v}^{2}}$

Calculate the first and second derivatives

Let $f \left(x\right) = {e}^{x} / \left(5 + {e}^{x}\right)$

$u \left(x\right) = {e}^{x}$, $\implies$, $u ' \left(x\right) = {e}^{x}$

$v \left(x\right) = 5 + {e}^{x}$, $\implies$, $v ' \left(x\right) = {e}^{x}$

Therefore,

$f ' \left(x\right) = \frac{{e}^{x} \left(5 + {e}^{x}\right) - {e}^{x} \cdot {e}^{x}}{5 + {e}^{x}} ^ 2 = \frac{5 {e}^{x}}{5 + {e}^{x}} ^ 2$

$\forall x \in \mathbb{R} , | , f ' \left(x\right) > 0$

No critical points.

Therefore,

The sign chart 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 , + \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}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a}$↗

Now, calculate the second derivative

$u \left(x\right) = 5 {e}^{x}$, $\implies$, $u ' \left(x\right) = 5 {e}^{x}$

$v \left(x\right) = {\left(5 + {e}^{x}\right)}^{2}$, $\implies$, $v ' \left(x\right) = 2 {e}^{x} \left(5 + {e}^{x}\right)$

$f ' ' \left(x\right) = \frac{5 {e}^{x} {\left(5 + {e}^{x}\right)}^{2} - 5 {e}^{x} \left(2 {e}^{x} \left(5 + {e}^{x}\right)\right)}{5 + {e}^{x}} ^ 4$

$= \frac{25 {e}^{x} + 5 {e}^{2 x} - 10 {e}^{2 x}}{{\left(5 + {e}^{x}\right)}^{3}}$

$= \frac{25 {e}^{x} - 5 {e}^{2 x}}{\left({\left(5 + {e}^{x}\right)}^{3}\right)}$

$= \frac{5 {e}^{x} \left(5 - {e}^{x}\right)}{\left({\left(5 + {e}^{x}\right)}^{3}\right)}$

The point of inflection is when $f ' ' \left(x\right) = 0$

$\implies$, $5 - {e}^{x} = 0$, $x = \ln 5$

We can make the chart

$\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}$$\left(- \infty , \ln 5\right)$$\textcolor{w h i t e}{a a a a}$$\left(\ln 5 , + \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}$$+$$\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}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a a a}$$\cap$

See the graph of the function

graph{e^x/(5+e^x) [-8.89, 8.89, -4.444, 4.445]}