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How to complete this derivative proof?

  1. Feb 16, 2015 #1
    1. The problem statement, all variables and given/known data

    Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$.

    Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then:

    $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) \tag 1$$



    2. Relevant equations

    Below

    3. The attempt at a solution


    **MY PROOF:**

    If $(1)$ is true then : $af(a) + bf(b) \ge af(a + b) + bf(a + b)$ , since it is given that: $af(a) \ge af(a+b)$ the objective is to prove: $f(b) \ge f(a + b)$

    Assume $f(b) < f(a + b) $

    $$\frac{f(a + b) - f(b)}{a} > 0 \tag2$$

    For some $x$ the **Mean Value Theorem** guarantees for $\exists x_1 \in (b, a+b)$ that $$f'(x_1) = \frac{f(a + b) - f(b)}{a} $$

    Hence, $(2)$ really means: $f'(x_1) > 0$ for some $x_1 \in (b, a+b)$

    Since, $f(a) \ge f(a + b)$:

    $$\frac{f(a + b) - f(a)}{b} \le 0 \tag3$$

    By the **Mean Value Theorem,** $\exists x_2 \in (a, a+b)$ such that $$f'(x_2) =\frac{f(a + b) - f(a)}{b}$$

    Hence $(3)$ really means, $$f'(x_2) \le 0$$

    1. Obviously, if $f'(x_2) = 0$ then $f''(x_2)$ is a max, which means $f''(x_2) = 0$
    2. Since $x_1 > x_2$ there $\exists x_3 \in (x_1, x_2)$ such that $f'(x_3) = 0$, which means $f''(x_3)$ is a max, meaning $f''(x_3) = 0$ contadictions.


    I just need help with 1 above, what if $x_2$ was a minimum?

    Thanks!
     
  2. jcsd
  3. Feb 16, 2015 #2

    RUber

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    If x2 is a minimum then f is constant if x2 is neither 0 or 2.
     
  4. Feb 16, 2015 #3

    SammyS

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    Use ## \small{\text{ ## ___ ## }} ## for delimiters for inline LaTeX.

    I fixed up some of it.
     
  5. Feb 17, 2015 #4
    @RUber, is my proof correct though?
     
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