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Math Help - How to solve this IVT problem??

  1. #1
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    How to solve this IVT problem??

    How do I solve this problem?

    Use IVT (Intermediate Value Theorem) to prove the following statement:

    2^x = bx has a solution if b>2

    Please help me solve this problem.

    Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?
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  2. #2
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    Quote Originally Posted by skboss View Post
    How do I solve this problem?

    Use IVT (Intermediate Value Theorem) to prove the following statement:

    2^x = bx has a solution if b>2

    Please help me solve this problem.

    Attempt: I know that IVT says that a continuous function has values in every single points, but if I don't the value in front of x how am I supposed to solve that?
    Let f(x)= 2^x- bx. Then f(1)= 2- b< 0. Whatever b is, there exist n such that 2^n> b. For that n, f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0.

    Now use the IVT
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Let f(x)= 2^x- bx. Then f(1)= 2- b< 0. Whatever b is, there exist n such that 2^n> b. For that n, f(2^n)= 2^{2^n}- b(2^n)= 4^n- 2^nb= 2^n(2^n- b)> 0.

    Now use the IVT

    Thank you very much, but I got lost after the 2nd step when u introduce n. If you can explain that, that would be great. Thanks again.
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  4. #4
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    Why part do you need explained?

    Do you understand why I wanted to be able to say that f(1)< 0 and that f(some number) > 0?

    Do you understand why 2^{2^n}= 4^n= (2^n)(2^n)?
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