Chapter 2.2, Problem 120E

To determine

ToDescribe: the error in the statement that graph of the function $f\left(x\right)=2x{\left(x-1\right)}^2{\left(x+3\right)}^3$ rises to the left and falls to the right.

Answer to Problem 120E

True.

Explanation of Solution

Given:

Function is given as $f\left(x\right)=1-x^3$ .

Calculation:

Simplifying the function.

  $\begin{array}{l}f\left(x\right)=2x{\left(x-1\right)}^2{\left(x+3\right)}^3\\ f\left(x\right)=2x\left(x^2+1-2x\right)\left(x^3+27+3x\left(3^2\right)+3x^2\cdot 3\right)\\ f\left(x\right)=\left(2x^3+2x-4x^2\right)\left(x^3+27+27x+9x^2\right)\\ f\left(x\right)=2x^3\times x^3+\mathrm{........}\\ f\left(x\right)=2x^6+\mathrm{........}\end{array}$

Term containing Leading coefficient is $2x^6$ .

Using leading coefficient test.

The graph of a polynomial function $f\left(x\right)=a_n{x}^n+\mathrm{.........}+a_{1}x+a_0$ rises or falls depends on the leading coefficient $a_n$ and degree of polynomial function.

Case III:

When $n$ is even and $a_n$ is positive then the graph rises to the left and falls to the right.

Here, for $f\left(x\right)=2x^6+\mathrm{.....}$ .

  $n=6$ is even and $a_n=2$ is positive the graph rises to the left and falls to the right.

So, the answer is true.