What are the values and types of the critical points, if any, of f(x)=300/(1+.03x^2)?

May 1, 2018

There is only a maximum point at $\left(0 , 300\right)$

Explanation:

We need

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

Compute the derivative of $f \left(x\right)$

$f \left(x\right) = \frac{300}{1 + 0.03 {x}^{2}}$

$f ' \left(x\right) = 300 \cdot \left(- \frac{1}{\left(1 + 0.03 {x}^{2}\right)}\right) \cdot 0.06 x$

$= - \frac{18 x}{1 + 0.03 {x}^{2}}$

The critical points are when

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

That is,

$- \frac{18 x}{1 + 0.03 {x}^{2}} = 0$

$x = 0$ as the denominator is $\forall x \in \mathbb{R} , \left(1 + 0.03 {x}^{2}\right) > 0$

Build a variation chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a}$$+ \infty$

$\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}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{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}$↗$\textcolor{w h i t e}{a a}$$300$$\textcolor{w h i t e}{a a a a}$↘

There is only a maximum point at $\left(0 , 300\right)$

graph{300/(1+0.03x^2) [-593.5, 640.5, -174, 443.4]}