Question

Use the Intermediate Value Theorem to show that the function $f\left(x\right)=-2x^2-3x+8$has at least one real zero on the interval$\left[0,4\right]$

Short answer

The function has a zero between 0 and 4.

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=-2x^2-3x+8$

Step 2. Explanation

Evaluate the function at 0 and 4.

$$\begin{gathered} f\left(0\right)=-2{\left(0\right)}^2-3\left(0\right)+8 \\ =0-0+8 \\ =8 \\ and \\ f\left(4\right)=-2{\left(4\right)}^2-3\left(4\right)+8 \\ =-32-12+8 \\ =-36 \end{gathered}$$

Thus, we have $f\left(0\right)>0andf\left(4\right)<0$and by intermediate value theorem, function has a zero between 0 and 4.