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let-a-b-gt-0-find-all-differentiable-function-f-0-0-such-that-f-a-x-bx-f-x-x-gt-0-




Question Number 204426 by universe last updated on 17/Feb/24
  let a , b >0  find all differentiable function     f:(0,∞)→(0,∞)  such that       f′((a/x))  =  ((bx)/(f(x)))    ,   ∀ x>0
$$\:\:\mathrm{let}\:\mathrm{a}\:,\:\mathrm{b}\:>\mathrm{0}\:\:\mathrm{find}\:\mathrm{all}\:\mathrm{differentiable}\:\mathrm{function} \\ $$$$\:\:\:\mathrm{f}:\left(\mathrm{0},\infty\right)\rightarrow\left(\mathrm{0},\infty\right)\:\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\mathrm{f}'\left(\frac{{a}}{\mathrm{x}}\right)\:\:=\:\:\frac{\mathrm{bx}}{\mathrm{f}\left(\mathrm{x}\right)}\:\:\:\:,\:\:\:\forall\:\mathrm{x}>\mathrm{0} \\ $$

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