# 009A Sample Midterm 2, Problem 2

The function  ${\displaystyle f(x)=3x^{7}-8x+2}$  is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that  ${\displaystyle f(x)}$  has a zero in the interval  ${\displaystyle [0,1].}$

Foundations:
What is a zero of the function  ${\displaystyle f(x)?}$
A zero is a value  ${\displaystyle c}$  such that  ${\displaystyle f(c)=0.}$

Solution:

(a)
Intermediate Value Theorem
If  ${\displaystyle f(x)}$  is continuous on a closed interval  ${\displaystyle [a,b]}$
and  ${\displaystyle c}$  is any number between  ${\displaystyle f(a)}$  and  ${\displaystyle f(b),}$

then there is at least one number  ${\displaystyle x}$  in the closed interval such that  ${\displaystyle f(x)=c.}$

(b)

Step 1:
First,  ${\displaystyle f(x)}$  is continuous on the interval  ${\displaystyle [0,1]}$  since  ${\displaystyle f(x)}$  is continuous everywhere.
Also,

${\displaystyle f(0)=2}$

and

${\displaystyle f(1)=3-8+2=-3.}$.

Step 2:
Since  ${\displaystyle y=0}$  is between  ${\displaystyle f(0)=2}$  and  ${\displaystyle f(1)=-3,}$
the Intermediate Value Theorem tells us that there is at least one number  ${\displaystyle x}$
such that  ${\displaystyle f(x)=0.}$
This means that  ${\displaystyle f(x)}$  has a zero in the interval  ${\displaystyle [0,1].}$