009A Sample Midterm 2, Problem 2

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

(a) State the Intermediate Value Theorem.

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

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

Solution:

(a)
Intermediate Value Theorem
If  $f(x)$  is continuous on a closed interval  $[a,b]$
and  $c$  is any number between  $f(a)$  and  $f(b),$
then there is at least one number  $x$  in the closed interval such that  $f(x)=c.$

(b)

Step 1:
First,  $f(x)$  is continuous on the interval  $[0,1]$  since  $f(x)$  is continuous everywhere.
Also,
$f(0)=2$
and         $f(1)=3-8+2=-3.$.

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

Final Answer:
(a)     See solution above.
(b)     See solution above.

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