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Question:

Let f(x) be a polynomial of degree four having extreme values at x=1 and x=2. If limx→0[1+f(x)/x2]=3, then f(2) is equal to

𕒼

𕒸

4

0

Solution:

Given that: limx→0[1+f(x)/x2]=3, f'(1)=0, f'(2)=0
To find: f(2)=?
Solution:
limx→0[1+f(x)/x2]=3
or, limx→0[x2+f(x)/x2]=3
since limit exits hence, x2+f(x)=ax4+bx3+3x2 ⇒f(x)=ax4+bx3+2x2
⇒f'(x)=4ax3+3bx2+4x
Also f'(x)=0 at x=1,2 ⇒a=1/2, b=-2
⇒f(x)=x4/2 -2x3+2x2
⇒f(2)=8-16+8=0
Hence, 0 is the correct option.