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Largest Interval for a Solution of y′=1+y² | JEE

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Question
The largest value of cc such that there exists a differentiable function h(x)h(x) for −c<x<c-c < x < c that is a solution of y′=1+y2y' = 1 + y^2 with h(0)=0h(0) = 0 is:
A2π2\pi
Bπ\pi
Cπ2\dfrac{\pi}{2}correct
Dπ4\dfrac{\pi}{4}
Solution
Separate and integrate:
dy1+y2=dx⇒tan⁡−1y=x+c1.\frac{dy}{1 + y^2} = dx \Rightarrow \tan^{-1}y = x + c_1.
 Apply h(0)=0h(0) = 0: tan⁡−10=0+c1⇒c1=0\tan^{-1}0 = 0 + c_1 \Rightarrow c_1 = 0, so y=tan⁡xy = \tan x. The function h(x)=tan⁡xh(x) = \tan x is differentiable only where tan⁡−1y=x\tan^{-1}y = x stays in the range (−π2,π2)\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right), i.e. −π2<x<π2-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}. Hence the largest such cc is π2\dfrac{\pi}{2}. Correct answer: (3)
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