Question

If \(f(x)=\tan ^{-1}\left[\frac{\log \left(\frac{e}{x^{2}}\right)}{\log \left(e x^{2}\right)}\right]+\tan ^{-1}\left[\frac{3+2 \log x}{1-6 \log x}\right]\) then the value of \(f^{\prime \prime}(x)\) is
(A) \(x^{2}\)
(B) \(x\)
(C) 1
(D) 0

Answer 1

Ans : (D)
 \(f(x)=\tan ^{-1}\left(\frac{1-2 \log x}{1+2 \log x}\right)+\tan ^{-1}\left(\frac{3+2 \log x}{1-6 \log x}\right)\)

let, \(2 \log x=\tan \theta\)
\(3=\tan \alpha\)
\(\therefore f(x)=\frac{\pi}{4}-\not+\alpha+\not\)
\(=\frac{\pi}{4}+\tan ^{-1}(3)=\) constant
\(\therefore \mathrm{f}^{\prime \prime}(\mathrm{x})=0\)

Answer 2

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