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Math Help - derivative proof

  1. #1
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    derivative proof

    show that if the second derivative of function f is greated than zero in the interval [a,b] then the first derivative of function f has atmost one zero in[a,b].and what iff the second derivative of function f is less than 0 throughout [a,b] instead

    in the proof do we have to generalise the ppolynomial or do directly by graph

    please help
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  2. #2
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    Re: derivative proof

    suppose by way of contradiction that f'(x) has two zeros in [a,b], say at x = c and x = d with c < d.

    then at some point k in (c,d) (which is contained in [a,b]), by rolle's therorem we have f"(k) = 0, contradicting that f"(x) > 0 on [a,b].

    the same proof works if f"(x) < 0 on [a,b].

    it is important that f"(x) > 0 and not f"(x) ≥ 0, for we might have f(x) = c (a constant), in which case f'(x) has infinitely many roots on any interval.
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  3. #3
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    Re: derivative proof

    thanks
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