Question

Let $f(x) = \log_e( \sin x), (0 < x < \pi)$ and $g(x) = \sin^{-1}(e^{-x}), (x \ge 0)$. If $\alpha$ is a positive real number such that $a = (fog)'(\alpha)$ and $b = (fog)(\alpha)$, then :

• A. $a \alpha^2 - b \alpha - a = 0$
• B. $a \alpha^2 - b \alpha - a = - 2 \alpha^2$
• C. $a \alpha^2 + b \alpha + a = 0$
• D. $a \alpha^2 - b \alpha - a = 1$

Answer 1

Answer: (D)

$fog(x) = (-x) \Rightarrow (fg(\alpha)) = -a = b$
$(fg(x))' = - 1 \Rightarrow (fg(\alpha))' = -1 = a$