Consider `f(x)=int_(-1)^(x)(e^((x-t)/(x-2-t))dt)/(x-2-t)^(2)`
Q. The y-intercept of tangent drawn to graph of `y=f(x)` at `x=-1` is

Options

A. 0
B. `(1)/(4)`
C. `(1)/(e)`
D. `4`

Answer

B

Explanation

Put `(1)/(x-2-t)=u`
`implies(1)/((x-2-t)^(2))dt=du`
`f(x)=int_((1)/(x-1))^(-(1)/(2))e.e^(2u)du`
`f(x)=(e)/(2).(e^(-1)-e^((2)/(x-1)))`
`f(x)=(1)/(2)-(e)/(2).e^((2)/(x-1))`
(1) `f(x)lt(1)/(2)` for all `xepsilonR`
`implies` Greatest integer in the Range `=0`
(2) `f^(')(x)=(e)/((x-1)^(2))e^((2)/(x-1))`
`f^(')(-1)=(1)/(4)(x+1)`
`impliesy` intercept`=(1)/(4)`