If $f:\mathbb R\to \mathbb R$ is differentiable, and $x^*\in \arg\max_x f(x)$, then $f'(x^*)=0$. Moreover, if $f$ is concave, or quasiconcave, then this condition is also sufficient, i.e. $x^*\in \arg\max_x f(x)$ iff $f(x^*)=0$.
Is there an analogous condition for random functions $f$?
That is, assume differentiable $f:\Theta\times \mathbb R\to \mathbb R$. Is it true under some conditions that
$$x^*\in \arg\max_x \mathbb E(f(\theta,x)) \quad\text{implies}\quad \mathbb E\left(\frac{\partial}{\partial x}f(\theta,x^*)\right)=0,$$
for a random variable $\theta$? And does "implies" similarly become "iff" when we assume $f$ is concave in $x$ (or in both $\theta$ and $x$)? Do we need to assume something about the distribution of $\theta$?