Let f(x) = loge(sin x), (0

Question

Let f(x) = loge(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 :

Accepted Answer

f(x) = ln(sin x), g(x) = sin--1 (e--x)

f(g(x)) = ln(sin(sin--1 e--x)) = -x

f(g($$\alpha $$)) = -- $$\alpha $$ = b

As f(g(x)) = -- x

$$ \therefore $$ (f(g(x)))' = -- 1

$$ \Rightarrow $$ (f(g($$\alpha $$)))' = -- 1 = a

$$ \therefore $$ b = -- $$\alpha $$,
a = -- 1

$$ \therefore $$ a$$\alpha $$2 - b$$\alpha $$ - a = - $$\alpha $$2 + $$\alpha $$2 + 1 = 1

Solutions · 1