If `("lim")_(tvecx)(e^tf(x)-e^xf(t))/((t-x)(f(x))^2)=2a n df(0)=1/2,` then find the value of `f^(prime)(0)dot` a) 4 (b) 2 (c) 0 (d) 1

Options

A. 4
B. 2
C. 0
D. 1

Explanation

`underset(trarrx)lim(e^(t)f(x)-e^(x)f(t))/((t-x)(f(x))^(2))=2`
`"or "underset(trarrx)lim(e^(t)f(x)-e^(x)f'(t))/(1cdot(f(x))^(2))=2" (Using L' Hospital Rule)"`
`"or "(e^(x)f(x)-e^(x)f'(x))/(1cdot(f(x))^(2))=2`
`"or "(d)/(dx)((e^(x))/(f(x)))=2`
`"or "(e^(x))/(f(x))=2x+c`
given f(0)=2
`"or "(1)/(f(0))=c`
`"or "c=2`
`"or "f(x)=(e^(x))/(2x+2)`
`therefore" "f'(x)=(e^(x)(2x+2)-2e^(x))/((2x+2)^(2))`
Hence, `(f'(0)=0`