Chapter 4.3, Problem 70E

To determine

To explain: The way in which graph changes as c decreases.

Explanation of Solution

Given information:

The equation is $P\left(x\right)=x^4+c{x}^3+x^2$ .

Calculation:

Consider the equation.

  $P\left(x\right)=x^4+c{x}^3+x^2$

Differentiate above equation.

  $P^{\prime }\left(x\right)=4x^3+3c{x}^2+2x$

Differentiate above equation.

  $P^{″}\left(x\right)=12x^2+6cx+2$

Substitute $y^{″}=0$ in above equation.

  $\begin{array}{l}12x^2+6cx+2=0\\ x=\frac{-6c\pm \sqrt{36c^2-96}}{24}\end{array}$

Since, the inflection point depends on discriminant.

  $\begin{array}{l}\sqrt{36c^2-96}=0\\ c=\pm \frac{2\sqrt{2}}{\sqrt{3}}\end{array}$

The graph is shown below.

  

As c decreases the function become strictly positive and strictly concave up.