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Math Help - Prove x^3 = cosx + 16

  1. #1
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    Prove x^3 = cosx + 16

    Prove x^3 = cosx + 16

    Not sure how to go about this one? Any suggestions?
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  2. #2
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    Quote Originally Posted by johnlaw View Post
    Prove x^3 = cosx + 16

    Not sure how to go about this one? Any suggestions?

    As is, this question makes no sense. Do you have any other information?
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  3. #3
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    the question reads:

    Prove that the equation x^3 = cosx + 16 has a solution

    that should help.. sorry for the miswording
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  4. #4
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    Quote Originally Posted by johnlaw View Post
    the question reads:

    Prove that the equation x^3 = cosx + 16 has a solution

    that should help.. sorry for the miswording
    That's better. You can prove there is a solution by graphing both y_1 = x^3 and y_2 =\cos x + 16 on the same set of axis. If they intersect you have proven there is a solution.

    Finding that solution is a new world of pain.
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  5. #5
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    Having graphed it, I see that there is an intersection at x slightly larger than 2. Using that, I can give a more formal proof by noting that 2^3= 8 while cos(2)+ 16 is greater than 15.5 and that 3^3= 27 while cos(3)+ 16 is only slightly larger than 15.

    That is, 2^3< cos(2)+ 16 while 3^3> cos(3)+ 16. Since x^3 and cos(x)+ 16 are continuous functions, there must be x between 2 and 3 such that x^3= cos(x)+ 16.
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  6. #6
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    Quote Originally Posted by johnlaw View Post
    the question reads:

    Prove that the equation x^3 = cosx + 16 has a solution

    that should help.. sorry for the miswording
    Use the Intermediate value Theorem on the function f(x) = x^3 - cos(x) - 16.
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