009A Sample Final 2, Problem 7

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Show that the equation    has exactly one real root.

1. 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  

2. Mean Value Theorem
        Suppose    is a function that satisfies the following:

         is continuous on the closed interval  

         is differentiable on the open interval  

       Then, there is a number    such that    and  


Step 1:  
First, we note that
Since    and   
there exists    with    such that
by the Intermediate Value Theorem.
Hence,    has at least one zero.
Step 2:  
Suppose that    has more than one zero.
So, there exist    such that
Then, by the Mean Value Theorem, there exists    with    such that
We have   
Therefore, it is impossible for    Hence,    has at most one zero.

Final Answer:  
        See solution above.

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