What is the condition for an even function to have a stationary point at $x=0$ and increasing in both other directions? Is there a condition (could be a differential equation), to classify certain functions that:  
a) are even, meaning $f(-x) = f(x),$
b) have a stationary point i.e $\frac{df}{dx} = 0$, specifically a minimum at $x=0$
c) are increasing, meaning they have a positive derivative increasing in the positive $x$ direction and due to being an even function, have a positive derivative while moving in the negative x direction as well.
Intuitively I am looking for a condition/differential equation to model all 2-D bowl like shapes. One simple example is a parabola:  
$$f(x) = x^2$$
 A: It's too general as a class of functions. We could say things like
$$f(x)=g(|x|)$$
for $g \colon [0,\infty)\to \mathbb{R}$ such that


*

*$g$ is differentiable

*$g'(0^+)=0$

*$g'(x)> 0$.


So we are talking of functions like $|x|^k$, $e^{|x|^k}$, $-\frac 1{1+|x|^k}$ for $k>1$, $\cosh(|x|)=\cosh(x)$ and many, many, many etceteras.
But actually, I'm just folding the graph alongside the $y$ axis to collapse your conditions for $x<0$ and $x>0$ into just one set of conditions. Nothing else.
A: Perhaps $f'$ is odd and increasing would work for you?
A: I'm assuming that you want $f$ to be a (at least) once differentiable function on the real numbers. In which case, here is one way to write your requirement:
$$f(x) = g(\vert x \vert) + c\quad \text{where} \quad c \in \mathbb R, \\\quad g\in C_1\left([0,\infty)\right), \; \;\; g'(0) = 0, \; \; \;g'(x) > 0 \text{ for } x > 0$$
Where the derivative of $g$ at $x=0$ is taken as the limit from the right.
In other words, $g$ is a strictly monotonically increasing function on $[0,\infty)$ except  for $x=0$ where $g'(0) = 0$. There is a one-to-one correspondence between these functions $g$ and the $f$ they produce (except for that $+c$ bit), so you may instead consider the class of functions $g$.
Examples of such functions:

