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Prove the following

Question:

Let $\mathrm{f}(\mathrm{x})=\log _{\mathrm{e}}(\sin \mathrm{x}),(0<\mathrm{x}<\pi)$ and $g(x)=\sin ^{-1}\left(e^{-x}\right),(x \geq 0)$. If $\alpha$ is a positive real number such that $\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)$ and $\mathrm{b}=(\mathrm{fog})(\alpha)$, then :

  1. $a \alpha^{2}-b \alpha-a=0$

  2. $a \alpha^{2}+b \alpha-a=-2 \alpha^{2}$

  3. $a \alpha^{2}+b \alpha+a=0$

  4. $a \alpha^{2}-b \alpha-a=1$


Correct Option: , 4

Solution:

fog $(x)=(-x) \Rightarrow(f g(\alpha))=-\alpha=b$

$(f g(x))^{\prime}=-1 \Rightarrow(f g(\alpha))^{\prime}=-1=a$

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