∵f(x) has extremum values at x=1 and x=2 ∵f′(1)=0 and f′(2)=0 As, f(x) is a polynomial of degree 4. Suppose f(x)=Ax4+Bx3+Cx2+Dx+E ∵
lim
x→0
(
f(x)
x2
+1)=3
⇒
lim
x→0
(
Ax4+Bx3+Cx2+Dx+E
x2
+1)=3
⇒
lim
x→0
(Ax2+Bx+C+
D
x
+
E
x2
+1)=3
As limit has finite value, so D=0 and E=0 Now A(0)2+B(0)+C+0+0+1=3 ⇒c+1=3⇒c=2 f′(x)=4Ax3+3Bx2+2Cx+D f′(1)=0⇒4A(1)+3B(1)+2C(1)+D=0 ⇒4A+3B=−4 ......(i) f′(2)=0⇒4A(8)+3B(4)+2C(2)+D=0 ⇒8A+3B=−2 ...........(i) From equations (i) and (ii), we get A=