Chapter 4.4, Problem 31E

To determine

To Describe:The several members of the family also investigate how maximum and minimum points and inflection points move when $c$ changes. You should also identify any transitional values of $c$ at which the basic shape of the curve changes

Answer to Problem 31E

The inflection points is at $x=-\frac{\sqrt{3}}{c},0,\frac{\sqrt{3}}{c}$

Maximum at $\left(\frac{1}{c},\frac{1}{2}\right)$ and absolute minimum at $\left(-\frac{1}{c},-\frac{1}{2}\right)$

Explanation of Solution

Given: $f(x)=\frac{cx}{1+c^2{x}^2}$

Given: $f(x)=\frac{cx}{1+c^2{x}^2}$

  $\begin{array}{l}f\text{'}(x)=\frac{c(c^2{x}^2-1)}{{(1+c^2{x}^2)}^2}\\ f\text{'}\text{'}(x)=\frac{2c^{3}x(c^2{x}^2-3)}{{(1+c^2{x}^2)}^3}\end{array}$

The graph of the function is

  

From the graph it is concluded for the $c=0$ function reduces to $y=0$

For

  $c\ne 0$ the function has the $x$ intercepts and the asymptotes is $y=0$

For

  $c<0$

The function has the reflection over the $x$ axis for $-c$

For

  $c>0$

The function have maximum at $\left(\frac{1}{c},\frac{1}{2}\right)$ and absolute minimum at $\left(-\frac{1}{c},-\frac{1}{2}\right)$

From the graph it is seen that as the value of $c$ increases, the extreme will shift towards $y$ axis.

The inflection points is at $x=-\frac{\sqrt{3}}{c},0,\frac{\sqrt{3}}{c}$