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Math Help - using Mean Value Theorem to solve an inequality

  1. #1
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    using Mean Value Theorem to solve an inequality

    I need to solve the following inequality using the MVT, not sure how to go about doing this:
    a^{t}b^{1-t}\leq ta+(1-t)b for t\in [0,1], a>0,b>0
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    I need to solve the following inequality using the MVT, not sure how to go about doing this:
    a^{t}b^{1-t}\leq ta+(1-t)b for t\in [0,1], a>0,b>0
    Where have you gotten stuck?
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  3. #3
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    Im not sure where to start
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    Im not sure where to start
    Most of these problems are solved by finding the correct interval [a,b] noting that f(b)-f(a)=(b-a)f'(c) for some c\in(a,b) and noticing that f'(c) is either increasing or decreasing on (a,b) so that (b-a)f'(a)\le f(b)-f(a)\le f'(b)(b-a) or the decreasing analogue. Is thist one of those?
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  5. #5
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    I am still thouroughly confused as to how to solve this one, all I have is that I should use the function f(x)=tx-x^t, but this hasnt really shown any light for me
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