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f-x-1-1-x-1-1-a-ax-ax-8-a-gt-0-x-gt-0-prove-1-lt-f-x-lt-2-

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Question Number 204640 by liuxinnan last updated on 24/Feb/24
f(x)=(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8))) a>0 x>0 prove 1<f(x)<2
$${f}\left({x}\right)=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}}}+\sqrt{\frac{{ax}}{{ax}+\mathrm{8}}} \\ $$$${a}>\mathrm{0}\:{x}>\mathrm{0} \\ $$$${prove}\:\mathrm{1}<{f}\left({x}\right)<\mathrm{2} \\ $$
Answered by lepuissantcedricjunior last updated on 26/Feb/24
x>0 a>0 x<x+1;a<a+1=>0<(x/(x+1))<x;0<(a/(1+a))<a =>0<(1/( (√(1+x))))<1 ;0<(1/( (√(1+a))))<1(1) 0<ax<ax+8 =>0<((ax)/(ax+8))<1=>0<(√((ax)/(ax+8)))<1(2) =>(1)+(2) =>0+0+0<(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8)))<1+1+1 =>0<(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8)))<3 =>0<1<(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8)))<2<3 1<(1/( (√(1+x))))+(1/( (√(1+a))))+(√((ax)/(ax+8)))<2 =>1<f(x)<2 .........prof cedric junior............
$$\boldsymbol{{x}}>\mathrm{0}\:\boldsymbol{{a}}>\mathrm{0} \\ $$$$\boldsymbol{{x}}<\boldsymbol{{x}}+\mathrm{1};{a}<{a}+\mathrm{1}=>\mathrm{0}<\frac{\boldsymbol{{x}}}{\boldsymbol{{x}}+\mathrm{1}}<\boldsymbol{{x}};\mathrm{0}<\frac{\boldsymbol{{a}}}{\mathrm{1}+\boldsymbol{{a}}}<\boldsymbol{{a}} \\ $$$$=>\mathrm{0}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{{x}}}}<\mathrm{1}\:;\mathrm{0}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{a}}}<\mathrm{1}\left(\mathrm{1}\right) \\ $$$$\mathrm{0}<{a}\boldsymbol{\mathrm{x}}<\boldsymbol{\mathrm{ax}}+\mathrm{8} \\ $$$$=>\mathrm{0}<\frac{\boldsymbol{\mathrm{ax}}}{\boldsymbol{\mathrm{ax}}+\mathrm{8}}<\mathrm{1}=>\mathrm{0}<\sqrt{\frac{{a}\boldsymbol{{x}}}{\boldsymbol{{ax}}+\mathrm{8}}}<\mathrm{1}\left(\mathrm{2}\right) \\ $$$$=>\left(\mathrm{1}\right)+\left(\mathrm{2}\right) \\ $$$$=>\mathrm{0}+\mathrm{0}+\mathrm{0}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{ax}}}{\boldsymbol{\mathrm{ax}}+\mathrm{8}}}<\mathrm{1}+\mathrm{1}+\mathrm{1} \\ $$$$=>\mathrm{0}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{ax}}}{\boldsymbol{\mathrm{ax}}+\mathrm{8}}}<\mathrm{3} \\ $$$$=>\mathrm{0}<\mathrm{1}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{ax}}}{\boldsymbol{\mathrm{ax}}+\mathrm{8}}}<\mathrm{2}<\mathrm{3} \\ $$$$\mathrm{1}<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{x}}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{ax}}}{\boldsymbol{\mathrm{ax}}+\mathrm{8}}}<\mathrm{2} \\ $$$$=>\mathrm{1}<\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)<\mathrm{2} \\ $$$$………{prof}\:{cedric}\:{junior}………… \\ $$$$ \\ $$$$ \\ $$

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