A basic question on second derivative of $f(x)$

Question

Is there any general shape of a curve for which $f''(x) >0$ for all $x$ ? the same question for $f''(x) < 0$ for all $x$

Answer 1

$f''(x)\gtrless0$ $\forall x$ $\Longrightarrow$ $f$ is strictly convex/concave. When $f$ is strictly convex, the graph of the function lies entirely above any tangent line to it, except for one point of tangency. Intuitively, the graph of a strictly convex function looks like an ever increasing, ever steeper roller coaster; or an ever decreasing, ever less steep roller coaster; or a roller coaster that is first decreasing and gets ever less steep, reaches a trough, and then becomes ever increasing and ever steeper.

By replacing words by their opposites appropriately, you get a characterization of the general shape of strictly concave functions.