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

It is given that f(x) is a fourth-degree polynomial such thatlimx→0 1+f(x)x2=3⇒limx→0 f(x)x2  is finite ⇒ f(x) has a repeated root atx=0.Let, f(x)=x2ax2+bx+c then.limx→0 1+f(x)x2=3⇒limx→0 1+ax2+bx+c=3⇒c+1=3⇒ c=2It is given that f(x) has extreme values at x=1 and x=2∴ f′(1)=0 and f′(2)=0⇒ 4a+3b+2c=0 32a+12b+4c=0∵f′(x)=4ax3+3bx2+2cx⇒ 4a+3b+4=0 and 32a+12b+8=0⇒ a=1/2,b=−2∴ f(x)=x212x2−2x+2⇒ f(2)=4(2−4+2)=0.