A continuous function f ( x ) on R arrow R satisfies the relation f( x )+f(2 x + y )+5 xy =f(3 x - y )+2 x 2+1 text for ∀ x , y ∈ R then which of the following hold(s) good?

Question

A continuous function f ( x ) on R arrow R satisfies the relation f( x )+f(2 x + y )+5 xy =f(3 x - y )+2 x 2+1 text for ∀ x , y ∈ R then which of the following hold(s) good? (A) f is many one (B) f has no minima (C) f is neither odd nor even (D) f is bounded

Tardigrade Admin · Accepted Answer

Let, 2 x+y=3 x-y ⇒ 2 y=x ⇒ y=(x/2) ∴ text put y =( x /2) f ( x )+ f ((5 x /2))+(5 x 2/2)= f ((5 x /2))+2 x 2+1 f ( x )=1-( x 2/2) ⇒ A , B

Q. A continuous function $f (x)$ on $R \to R$ satisfies the relation $f (x) + f (2 x + y) + 5 x y = f (3 x - y) + 2 x^{2} + 1 for \forall x, y \in R$ then which of the following hold(s) good?

f is many one

f has no minima

$f$ is neither odd nor even

$f$ is bounded

Let, $2 x + y = 3 x - y \Rightarrow 2 y = x \Rightarrow y = \frac{x}{2}$
$∴ put y = \frac{x}{2}$
$f (x) + f (\frac{5 x}{2}) + \frac{5 x ^{2}}{2} = f (\frac{5 x}{2}) + 2 x^{2} + 1$
$f (x) = 1 - \frac{x ^{2}}{2} \Rightarrow A, B$

Q. A continuous function f(x) on R→R satisfies the relation f(x)+f(2x+y)+5xy=f(3x−y)+2x2+1 for ∀x,y∈R then which of the following hold(s) good?

f is many one

f has no minima

f is neither odd nor even

f is bounded

Let, 2x+y=3x−y⇒2y=x⇒y=2x​ ∴ put y=2x​ f(x)+f(25x​)+25x2​=f(25x​)+2x2+1 f(x)=1−2x2​⇒A,B

Q. A continuous function $f (x)$ on $R \to R$ satisfies the relation $f (x) + f (2 x + y) + 5 x y = f (3 x - y) + 2 x^{2} + 1 for \forall x, y \in R$ then which of the following hold(s) good?

$f$ is neither odd nor even

$f$ is bounded

Let, $2 x + y = 3 x - y \Rightarrow 2 y = x \Rightarrow y = \frac{x}{2}$
$∴ put y = \frac{x}{2}$
$f (x) + f (\frac{5 x}{2}) + \frac{5 x ^{2}}{2} = f (\frac{5 x}{2}) + 2 x^{2} + 1$
$f (x) = 1 - \frac{x ^{2}}{2} \Rightarrow A, B$