Chapter 2.3, Problem 69E

Solution

To determine whether the statement is true or false:

The graph of the function $f\left(x\right)=x^3-x^2-2$ crosses the $x$ -axis in between $x=1$ and $x=2$ .

The statement that graph of the function $f\left(x\right)=x^3-x^2-2$ crosses the $x$ -axis in between $x=1$ and $x=2$ is true.

Given the function:

  $f\left(x\right)=x^3-x^2-2$

Explanation:

The function $f\left(x\right)=x^3-x^2-2$ is a polynomial in $x$ .

Hence, it is continuous everywhere.

Now, find $f\left(1\right)$ :

  $\begin{array}{l}f\left(1\right)=1^3-1^2-2\\ f\left(1\right)=-2\end{array}$ .

That is, $f\left(1\right)$ is negative.

Now, find $f\left(2\right)$ :

  $\begin{array}{l}f\left(1\right)=2^3-2^2-2\\ f\left(1\right)=2\end{array}$ .

That is, $f\left(2\right)$ is positive.

Now, since $f\left(x\right)=x^3-x^2-2$ is continuous, it assumes all the values in between $f\left(1\right)=-2$ and $f\left(2\right)=-2$ when $x$ varies from $1$ to $2$ .

In particular, $f\left(x\right)$ will assume $0$ for some $a$ that lies in between $1$ and $2$ .

That is, $f\left(x\right)$ crosses the $x$ -axis somewhere at $x=a$ in between $x=1$ and $x=2$ .

That is, the graph of the function $f\left(x\right)=x^3-x^2-2$ crosses the $x$ -axis somewhere in between $x=1$ and $x=2$ .

Hence, the statement that graph of the function $f\left(x\right)=x^3-x^2-2$ crosses the $x$ -axis in between $x=1$ and $x=2$ is true.