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f-x-1-1-x-1-1-a-ax-ax-8-x-0-a-0-show-1-lt-f-x-lt-2-




Question Number 138316 by Ñï= last updated on 12/Apr/21
f(x)=(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8)))        ,x∈(0,∞)  ∀ a∈(0,∞),show 1<f(x)<2.
$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}}}+\sqrt{\frac{{ax}}{{ax}+\mathrm{8}}}\:\:\:\:\:\:\:\:,{x}\in\left(\mathrm{0},\infty\right) \\ $$$$\forall\:{a}\in\left(\mathrm{0},\infty\right),{show}\:\mathrm{1}<{f}\left({x}\right)<\mathrm{2}. \\ $$

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