Let f: R R be a polynomial function defined by f(x)=(a-2) x^3+(b+1) x^2+c x+d . If f(x+y)=f(x)+f(y)+4 x y-5 for all x, y R and f^ (1)=8 , then the value of a+b+c+d is

Step-by-step solution

Substitute x=0, y=0 into the given functional equation: f(0) = f(0) + f(0) + 0 - 5 f(0) = 5 Since f(x) = (a-2)x^3 + (b+1)x^2 + cx + d , we have f(0) = d d = 5 . Now, evaluate f(x+y) - f(x) - f(y) : (a-2)[(x+y)^3 - x^3 - y^3] + (b+1)[(x+y)^2 - x^2 - y^2] + c[(x+y) - x - y] + d - 2d = (a-2)(3x^2y + 3xy^2) + (b+1)(2xy) - d This must equal 4xy - 5 for all x, y R . Comparing coefficients: For the cubic terms ( x^2y and xy^2 ), a - 2 = 0 a = 2 . For the xy term, 2(b+1) = 4 b+1 = 2 b = 1 . The constant term is -d = -5 d =