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JEE MATHEMATICS MCQ: INCREASING & DECREASING

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Analyzing +4 | -1
Let g(x)=f(x)+f(1−x)g(x)=f(x)+f(1-x) and f′′(x)>0,x∈(0,1)f^{\prime \prime}(x)>0, x \in(0,1). If gg is decreasing in the interval (0,α)(0, \alpha) and increasing in the interval (α,1)(\alpha, 1), then tan⁡−1(2α)+\tan ^{-1}(2 \alpha)+ tan⁡−1(1α)+tan⁡−1(α+1α)\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right) is equal to
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Correct option: A
π\pi
Option A (π\pi) is correct because finding the critical point of g(x) leads to α=1/2\alpha = 1/2, and substituting this value into the expression yields tan⁡−1(1)+tan⁡−1(2)+tan⁡−1(3)\tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3), which equals π\pi.

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