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Suppose |f^(prime)(x)f(x)f^(x)f^(prime)(...

Suppose `|f^(prime)(x)f(x)f^(x)f^(prime)(x)|=0"w h e r ef"("x")` is continuously differentiable function with `f^(prime)(x)!=0` and satisfies `f(0)=1` and `f^(prime)(0=2)` then `(lim)_(xvec0)(f(x)-1)/x` is `1//2` b. `1` c. `2` d. `0`

A

1

B

2

C

`1//2`

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
`|(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`
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