Let $f:\mathbb{R}->\mathbb{R}$ be a function with the property that for every $y \in\mathbb{R}$ the value of the expression $\sup_{x \in \mathbb{R}}[xy-f(x)]$ is finite. Define
$g(y)=\sup_{x \in \mathbb{R} }[xy-f(x)]$ for $y \in \mathbb{R}$. Then
a. g is even if f is even
b. f must satisfy $\lim_{|x|->\infty}\frac{f(x)}{x}=\infty$
c. g is odd if f is even
d. f must satisfy $\lim_{|x|->\infty}\frac{f(x)}{x}=-\infty$
I don't even know how to start. I tried to find suitable f and g and eliminate options. But this idea is also not working. This problem is really difficult for me. Kindly help me.