Question

A continuous function $f ( x )$ on $R \rightarrow R$ satisfies the relation $f( x )+f(2 x + y )+5 xy =f(3 x - y )+2 x ^2+1 \text { for } \forall x , y \in R$ 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

Answer 1

Answer: (A), (B)

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