Question

Use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval.

$f\left(x\right)=8x^4-4x^3-2x-1;[0,1]$

Short answer

The given function has a zero between 0 and 1

Step-by-step solution

Step 1. Given Information

The given function and the interval is$f\left(x\right)=8x^4-4x^3-2x-1;[0,1]$

Step 2. Calculation 

We will evaluate the function at 0 and 1.

$$\begin{gathered} f\left(0\right)=8{\left(0\right)}^4-4{\left(0\right)}^3-2\left(0\right)-1 \\ =-1 \\ f\left(1\right)=8{\left(1\right)}^4-4{\left(1\right)}^3-2\left(1\right)-1 \\ =1 \end{gathered}$$

Since,$f\left(0\right)<0andf\left(1\right)>0andf\left(0\right)andf\left(1\right)$are of opposite sign, it follows from the intermediate value theorem that the function has a zero between 0 and 1.