Let `f (x) = log _e (sinx ), ( 0 lt x lt 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

Options

A. `aalpha ^(2) - b alpha - a = 0`
B. `a alpha ^(2) - b alpha - a = 1 `
C. `a alpha ^(2) + b alpha - a = - 2 alpha ^(2)`
D. `a alpha ^(2) + b alpha + a = 0`

Answer

B

Explanation

Given functions, `f(x) = log_e ( sin x ), (0 lt x lt pi) and g(x) = sin ^(-1) (e^(-x)), x ge 0`.
Now, `fog(x) = f(g(x))= f( sin^(-1)(e^(-x)))`
`= log_e (sin(sin ^(-1)(e^(-x))))`
`= log_e (e^(-x))" " {because sin ( sin ^(-1)x ) = x, if x in [-1, 1]}`
` = -x " "` ...(i)
and `" " ( fog)' (x) = (d)/(dx) (-x) =-1" "`... (ii)
According to the equation,
`because a = (fog)' (alpha) =-1" " `[from Eq. (ii)]
and `b = (fog)(alpha = - (alpha)" " `[from Eq. (i)]
for a positive real value `'alpha'`.
Since, the value of `a =-1 and b =-alpha`, satisfy the quadratic equation ( from the given options)
`" " a alpha^(2) - b alpha - alpha =1`.