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Question

Let f(x) be a polynomial of degree four having extreme value at x=1 and x=2.If limx0(1+f(x)x2)=3,then f(2) is equal to ?

A
0
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B
4
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C
8
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D
49
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Solution

The correct option is D 49
Given f(1)=f(2)=0
limx0(1+f(x)x2)=3
limx0f(x)x2=2f(0)=0 ( for applying L-hospital rule)
Apply L-Hospital Rule
f′′(0)2=2f′′(0)=4
Let f(x)=(x1)(x2)(xa)
f′′(x)=(x1)(x2)+(x2)(xa)+(x1)(xa)
f′′(0)=2+2a+a=4a=23
f(x)=(x1)(x2)(x23)=x3113x2+4x43
f(x)=x44119x3+2x243x+d
f(0)=0d=0
f(2)=4889+883=49

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