Suppose `|(f'(x),f(x)),(f''(x),f'(x))|=0` where f(x) is continuously differentiable function with `f'(x)ne0` and satisfies f(0) = 1 and f'(0) = 2 then `lim_(xrarr0) (f(x)-1)/(x)` is

Options

A. 1
B. 2
C. `1//2`
D. 0

Answer

B

Explanation

`|(f'(x),f(x)),(f''(x),f'(x))|=0`
`rArr" "f'(x)f'(x)-f(x).f''(x)=0`
`rArr" "((f(x))^(2)-f(x)f''(x))/((f'(x))^(2))=0`
`rArr" "(d)/(dx)[(f(x))/(f'(x))]=0`
`rArr" "(f(x))/(f'(x))=C" (i)"`
Put `x=0, (f(0))/(f'(0))=C rArr C=(1)/(2)`
Hence `(f(x))/(f'(x))=(1)/(2)`
From (i), `2f(x)=f'(x)`
`therefore" "(f'(x))/(f(x))=2`
`therefore" "(d)/(dx)(logf(x))=2`
`therefore" "log(f(x))=2x+k`
Putting x = 0, we get k = 0
`rArr" "f(x)=e^(2x)`
`rArr" "underset(xrarr0)(lim)(e^(2x)-1)/(x)=2`