92
52
32
12
Given it has extremum values at x=1 and x=2 ⇒f'(1)=0 and f'(2)=0
Given f(x) is a fourth degree polynomial
Let f(x)=ax4+bx3+cx2+dx+c=0
Given limx→0(f(x)/(x2+1))=3
limx→0(ax4+bx3+cx2+dx+e)/(x2+1)=3
limx→0(ax2+bx+c+dx+ex2+1)=3
For limit to have finite value (in this case 3) value of 'd' and 'e' must be 0 ⇒d=0 e=0
Substituting x=0 in limit ; ⇒c+1=3 ⇒c=2
f'(x)=4ax3+3bx2+2cx+d
Applying f'(1)=0, f'(2)=0
4a(1)+3b(1)+2c(1)+d=0 ⇒4a+3b+4=0
4a(8)+3b(4)+2c(2)+d=0 ⇒32a+12b+8=0
Solving two equations, we get a=1/2 and b=-6
f(x)=x4/2 -6x3+2x2
f(-1)=(-1)4/2 -6(-1)3+2(-1)2
Hence f(x)=92
Therefore correct option is 'D'