Chapter 4, Problem 10CT

To determine

To prove: That the polynomial function $f(x)=-2x^2-3x+8$ has a real zero in the interval $\left[0,4\right]$ by using the Intermediate Value Theorem.

Explanation of Solution

Given information:

The polynomial function $f(x)=-2x^2-3x+8$ ; $\left[0,4\right]$ .

Formula used:

Intermediate Value Theorem:

Let $f$ denote a polynomial function. If $a<b$ and if $f\left(a\right)$ and $f\left(b\right)$ are of opposite sign, there

is at least one real zero of $f$ between $a$ and $b$ .

Proof:

Evaluate the polynomial function $f(x)=-2x^2-3x+8$ by substituting $x=0$ and $x=4$ in it.

  $f(0)=-2{\left(0\right)}^2-3\left(0\right)+8$

  $\Rightarrow f(0)=8$

And $f(4)=-2{\left(4\right)}^2-3\left(4\right)+8$

  $\Rightarrow f(4)=-36$

Here, $f(0)>0$ and $f(4)<0$ .

Therefore, by using the Intermediate Value theorem, the polynomial function $f(x)=-2x^2-3x+8$ has at least one real zero in the interval $\left[0,4\right]$ .

Hence, proved.