009A Sample Midterm 2, Problem 2

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The function    is a polynomial and therefore continuous everywhere.

(a) State the Intermediate Value Theorem.

(b) Use the Intermediate Value Theorem to show that    has a zero in the interval  


Foundations:  
What is a zero of the function  
        A zero is a value    such that  


Solution:

(a)  
Intermediate Value Theorem
        If    is continuous on a closed interval  
        and    is any number between    and  

        then there is at least one number    in the closed interval such that  

(b)

Step 1:  
First,    is continuous on the interval    since    is continuous everywhere.
Also,

       

and

        .

Step 2:  
Since    is between    and  
the Intermediate Value Theorem tells us that there is at least one number  
such that  
This means that    has a zero in the interval  


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

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