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Question Number 39381 by rahul 19 last updated on 05/Jul/18

Answered by ajfour last updated on 06/Jul/18

$$f{}^{\prime }\left(x\right)⩾{\left[f\left(x\right)\right]}^3+{\left[f\left(x\right)\right]}^{-1}$$$$andf\left(0\right)=1,f\left(a\right)=3^{1/4}$$$$\Rightarrow {\int }_1^y\frac{dy}{y^3+\frac{1}{y}}⩾{\int }_0^{x}dx$$$$\Rightarrow \frac{1}{2}{\int }_1^y\frac{2ydy}{y^4+1}⩾x$$$$or{\tan }^{-1}{y}^2-{\tan }^{-1}1⩾2x$$$$forx=a,y^2=\sqrt{3}$$$$\Rightarrow 2a⩽\frac{\pi }{3}-\frac{\pi }{4}$$$$\frac{216a}{\pi }⩽9.$$

Commented by rahul 19 last updated on 06/Jul/18

from where dx come ? why it's less than than dy wala integral (Step 1)

Commented by ajfour last updated on 06/Jul/18

$$f\left(x\right)=y$$$$f{}^{\prime }\left(x\right)=\frac{dy}{dx}$$$$itisgiven\frac{dy}{dx}⩾y^3+\frac{1}{y}$$$$\Rightarrow \frac{dy}{y^3+\frac{1}{y}}⩾dx$$$$nowweneedtointegrate..$$

Commented by rahul 19 last updated on 06/Jul/18

Thank you Sir ! ��������

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