Chapter 4.4, Problem 32E

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 32E

  

Explanation of Solution

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

Given: $f(x)=\frac{1}{(1-x^2)+c{x}^2}=\frac{1}{1+x^4+a{x}^2}$

  $a=c-2$

  $f(x)$ will have a maximum when $g(x)$ will have minimum

Where $g(x)=x^4+a{x}^2+1$

The graph of the function is

  

For

  $a<0,a^2-4<0$

  $\begin{array}{l}a=-1\\ c=1\end{array}$

For

  $\begin{array}{l}a\le 0,a^2-4=0\\ a=-2\\ c=0\end{array}$

For

  $\begin{array}{l}a<0,a^2-4>0\\ a=-8\\ c=-6\end{array}$

For

  $\begin{array}{l}a>0,a^2-4<0\\ a=1\\ c=3\end{array}$

For

  $\begin{array}{l}a>0,a^2-4>0\\ a=8\\ c=10\end{array}$