I see in wolfram that a function $f$ is pseudo convex if it satisfies following,
$\nabla f(x)\cdot (y-x) ≥ 0 \Rightarrow f(y) ≥ f(x) $
My question is, with this definition, how come $g(x)=x^3$ is not pseudoconvex, while $h(x)=x+x^3$ is pseudo convex?
$g(x) = x^3$
$\\ \\ \\$
$h(x) = x+x^3$
We have $h'(x) > 0$ for any $x \in \Bbb R$, so to check pseudoconvexity it suffices to check that: $y-x \ge 0 \implies h(y) \ge h(x)$, i.e. just that $h$ is increasing, which is obviously the case.
The problem for $g$ is that $g'(0) = 0$, so we can choose, say, $x = 0$ and $y = -1$ to see that the implication doesn't hold.