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If f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3...

If `f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3(98))…(x-100)^(100),` then the value of `(f'(101))/(f(101))` is

A

5050

B

2575

C

3030

D

1250

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=prod_(i=l)^(100)(x-i)^(i(101-i))`
Taking log both sides, we get
`log(f(x))=sum_(i=l)^(100)i(101-i)log(x-i)`
Differentiating w.r.t. x, we get
`(1)/(f(x)).f'(x)=sum_(i=l)^(100)(i(101-i))/((x-i))`
`rArr" "(f'(100))/(f(101))=sum_(i=l)^(100)((101-i)/(101-i))=5050`
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