Let c, k ∈ R. If f(x)=(c+1) x2+(1-c2) x+2 k and f(x+y)=f(x)+f(y)-x y, for all x, y ∈ R, then the value of |2(f(1)+f(2)+f(3)+ ldots ldots+f(20))| is equal to

Question

Let c, k ∈ R. If f(x)=(c+1) x2+(1-c2) x+2 k and f(x+y)=f(x)+f(y)-x y, for all x, y ∈ R, then the value of |2(f(1)+f(2)+f(3)+ ldots ldots+f(20))| is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

beginarrayl f(x)=(c+1) x2+(1-c2) x+2 k f(x+y)=f(x)+f(y)-x y ∀ x y ∈ R lim y arrow 0 (f(x+y)-f(x)/y)= lim y arrow 0 (f(y)-x y/y) f(x)=-(1/2) x2+f prime(0) ⋅ x+λ text but f(0)=0 ∴(x)=-(1/2) x2+(1-c2) ⋅ x ∴ text as f prime(0)=1-c2 endarray Comparing equation (1) and (2) We obtain, c=-3 / 2 ∴ f ( x )=-(1/2) x 2-(5/4) x Now |2 ∑x=120 f(x)|=∑x=120 x2+(5/2) ⋅ ∑x=120 x =2870+525 =3395

Q. Let c,k∈R. If f(x)=(c+1)x2+(1−c2)x+2k and f(x+y)=f(x)+f(y)−xy, for all x,y∈R, then the value of ∣2(f(1)+f(2)+f(3)+……+f(20))∣ is equal to _______

Q. Let $c, k \in R$ . If $f (x) = (c + 1) x^{2} + (1 - c^{2}) x + 2 k$ and $f (x + y) = f (x) + f (y) - x y$ , for all $x, y \in R$ , then the value of $∣2 (f (1) + f (2) + f (3) + \dots\dots + f (20)) ∣$ is equal to _______