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Math Help - derivatives

  1. #1
    Junior Member
    Joined
    Jun 2009
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    derivatives

    Hi everyone.

    I need to prove something and can't seem to make
    heads or tails on how to start.

    - f is a function in which f(0) = 0 and f``(x)<=0 for every x.
    if a,b>=0 prove that f(a+b)<=f(a)+f(b).

    tanks in advance,
    i hope i'm clear in my description.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    1. Since f " < 0 then f ' (x) is decreasing

    2. define h(x) = f(x+b) -f(x)

    3. h ' (x) = f '(x + b) - f ' (x) < 0 since f ' is decreasing

    4. h(a) = f(a+b)- f(a)

    h(0) = f(b)

    h(a)- h(0) = f(a+b) - [ f(a) + f(b)]

    5. by the MVT there is a number c in (0,a) st

    a* h'(c) = h(a) -h(0) = f(a+b) - [ f(a) + f(b)] < 0

    the result follows.
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  3. #3
    Junior Member
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    thanks!
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