Let f be a differentiable function such that f ( x + y )= f ( x )+ f ( y )+2 xy -1, for all real x and y. If f prime(0)= cos α, then ∀ x ∈ R.

Question

Let f be a differentiable function such that f ( x + y )= f ( x )+ f ( y )+2 xy -1, for all real x and y. If f prime(0)= cos α, then ∀ x ∈ R. (A) f (x)<0 (B) f(x)=0 (C)  f ( x )>0 (D) f(x)=x

Tardigrade Admin · Accepted Answer

f(x+y)=f(x)+f(y)+2 x y-1, [ Put x =0, y =0, ∴ f (0)=1] Partial differentiation w. r. to ' x ' f prime( x + y )= f prime( x )+2( y ) Replace y arrow x, x arrow 0 f prime( x )= f prime(0)+2 x f prime( x )=2 x + cos α Integral, f(x)=x2+x cos α+c f (0)=1 ⇒ 1= c f ( x )= x 2+ x cos x +1 D = cos 2 α-4<0,( text coefficient of x 2>0. text ) ∴ f ( x )>0 .

Q. Let $f$ be a differentiable function such that $f (x + y) = f (x) + f (y) + 2 x y - 1$ , for all real $x$ and $y$ . If $f^{'} (0) = cos α$ , then $\forall x \in R$ .

f $(x) < 0$

$f (x) = 0$

$f (x) > 0$

$f (x) = x$

Q. Let f be a differentiable function such that f(x+y)=f(x)+f(y)+2xy−1, for all real x and y. If f′(0)=cosα, then ∀x∈R.

f (x)<0

f(x)=0

f(x)>0

f(x)=x

f(x+y)=f(x)+f(y)+2xy−1, [ Put x=0,y=0, ∴f(0)=1] Partial differentiation w. r. to ' x ' f′(x+y)=f′(x)+2(y) Replace y→x,x→0 f′(x)=f′(0)+2x f′(x)=2x+cosα Integral, f(x)=x2+xcosα+c f(0)=1⇒1=c f(x)=x2+xcosx+1 D=cos2α−4<0,( coefficient of x2>0 ) ∴f(x)>0.

Q. Let $f$ be a differentiable function such that $f (x + y) = f (x) + f (y) + 2 x y - 1$ , for all real $x$ and $y$ . If $f^{'} (0) = cos α$ , then $\forall x \in R$ .

f $(x) < 0$

$f (x) = 0$

$f (x) > 0$

$f (x) = x$