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if-2x-1-x-1-2-find-the-value-8x-1-x-




Question Number 181707 by amin96 last updated on 28/Nov/22
if     2x+(1/( (√x)))=(1/2)  find the value   8x+(1/( (√x)))=?
$$\boldsymbol{{if}}\:\:\:\:\:\mathrm{2}\boldsymbol{{x}}+\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{x}}}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\:\:\mathrm{8}\boldsymbol{{x}}+\frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{x}}}}=? \\ $$
Answered by mr W last updated on 29/Nov/22
t=(√x)>0  f(t)=2t^2 +(1/t)  (df/dt)=4t−(1/t^2 )=0 ⇒t=(1/( (4)^(1/3) ))  f_(min) =(1/( (2)^(1/3) ))+(4)^(1/3) ≈2.381 >(1/2)  ⇒2x+(1/( (√x))) =(1/2) is not possible for x∈R.
$${t}=\sqrt{{x}}>\mathrm{0} \\ $$$${f}\left({t}\right)=\mathrm{2}{t}^{\mathrm{2}} +\frac{\mathrm{1}}{{t}} \\ $$$$\frac{{df}}{{dt}}=\mathrm{4}{t}−\frac{\mathrm{1}}{{t}^{\mathrm{2}} }=\mathrm{0}\:\Rightarrow{t}=\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{4}}} \\ $$$${f}_{{min}} =\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{2}}}+\sqrt[{\mathrm{3}}]{\mathrm{4}}\approx\mathrm{2}.\mathrm{381}\:>\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{2}{x}+\frac{\mathrm{1}}{\:\sqrt{{x}}}\:=\frac{\mathrm{1}}{\mathrm{2}}\:{is}\:{not}\:{possible}\:{for}\:{x}\in{R}. \\ $$

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