Chapter 4, Problem 23RE

To determine

To show: The polynomial function $f(x)=3x^3-x-1$ has a zero in the given interval $\left[0,1\right]$ .

Answer to Problem 23RE

The polynomial function has at least one zero between and 0 and 1 .

Explanation of Solution

Given information:

The polynomial function $3x^3-x-1$ .

The interval $\left[0,1\right]$ .

Formula used:

Intermediate Value Theorem:

Let f be a continuous function. If $a<b$ and if $f(a)$ and $f(b)$ are of opposite sign, the f has at least one zero between a and b .

Calculation:

Consider ,

  $f(x)=3x^3-x-1;\left[0,1\right]$

Since it is a polynomial function . Hence, it is a continuous function.

  $=a=0;b=1$

This implies $0<1$ .

  $\begin{array}{l}=f(a)=3a^3-a-1\\ =f(0)=3(0)-0-1\end{array}$

  $=-1$

and

  $\begin{array}{l}=f(b)=3b^3-b-1\\ =f(1)=3(1)-1-1\end{array}$

  $\begin{array}{l}=3-1-1\\ =3-2\\ =1\end{array}$

Therefore, $f(a)$ and $f(b)$ have opposite sign.

Hence, f has at least one zero between 0 and 1.