calculas

let f : $\mathbb{R}\implies\mathbb{R}$ be a monotonic function for which there exist a,b , c, d $\in$ with a $\neq$ 0 , c $\neq$ 0 such that for all x the following equalities hold :
$\int_{x}^{x+\sqrt{3}}$f(t)dt = ax+b and $\int_{x}^{x+\sqrt{2}}$f(t)dt =cx+d.
Prove that f is a linear function , i.e, f(x)=mx+n for some m,n$\in $$\mathbb{R}$

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One thought on “calculas

  1. Solution :(AOPS)
    #6
    VPM Thursday at 6:27 PM
    neelanjan wrote:
    let f : $\mathbb{R}\implies\mathbb{R}$ be a monotonic function for which there exist a,b , c, d $\in$ with a $\neq$ 0 , c $\neq$ 0 such that for all x the following equalities hold :
    $\int_{x}^{x+\sqrt{3}}$f(t)dt = ax+b and $\int_{x}^{x+\sqrt{2}}$f(t)dt =cx+d.
    Prove that f is a linear function , i.e, f(x)=mx+n for some m,n$\in $$\mathbb{R}$

    $\int_{x}^{x+\sqrt{3}}f(t)dt=ax+b;\int_{x}^{x+\sqrt{2}}f(t)dt =cx+d$

    $\\ \Rightarrow \left ( \int_{x}^{x+\sqrt{3}}f(t)dt \right )’=(ax+b)’ \Leftrightarrow f(x+\sqrt{3})-f(x)=a;f(x+\sqrt{2})-f(x)=c$

    $f(x+\sqrt{3})-f(x)=a$ g(x)=a $ \Rightarrow g(x) $ is diffrenable$ \Rightarrow \exists $diffrenable $ (f(x+\sqrt{3})-f(x))$

    Diffrenable is lim ;$ \lim {(f(x+\sqrt{3})-f(x))} =\lim{f(x+\sqrt{3})}+\lim{f(x)} \Rightarrow \exists f'(x) $

    $\\ \Leftrightarrow f'(x+\sqrt{3})=f'(x);f'(x+\sqrt{2})=f'(x)$

    $\\ g(x)=f'(x) \Rightarrow g(x+\sqrt{2})=g(x);g(x+\sqrt{3})=g(x) \forall x \in R $

    $\\ \sqrt{3};\sqrt{2}$ are period of $g(x) \Rightarrow \sqrt{3}=n\sqrt{2} (n\in Z) $

    It is nonsense $\Rightarrow g(x)=const \Rightarrow f'(x)=c \Rightarrow f(x)=mx+n$

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