Let f(x) be a polynomial of degree 4 having extreme values at x=1 and x=2 . If _ x 0 ( f(x) x^2 +1 )=3 then f(-1) is equal to

Step-by-step solution

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)=A x^4+B x^3+C x^2+D x+E aligned & _ x 0 ( f(x) x^2 +1 )=3 & _ x 0 ( A x^4+B x^3+C x^2+D x+E x^2 +1 )=3 & _ x 0 (A x^2+B x+C+ D x + E x^2 +1 )=3 aligned As limit has finite value, so D=0 and E=0 Now A(0)^2+B(0)+C+0+0+1=3 aligned & c+1=3 c=2 &f^ (x)=4 A x^3+3 B x^2+2 C x+D &f^ (1)=0 4 A(1)+3 B(1)+2 C(1)+D=0 aligned aligned & 4 A+3 B=-4 f^ (2)=0 & 4 A(8)+3 B(4)+2 C(2)+D=0 & 8 A+3 B=-2 aligned From equation