# Stationary conditions for intersection

$$\mbox{min} f(x)$$
$$x\in C$$
Let $$f$$ be $$C^1$$ function over a closed and convex set $$C$$ . then $$x^*$$ is called a stationary point of (P) if $$\nabla f(x^*)(x-x^*)\geq0$$ for any $$x\in C$$
Assume that we know the stationary condition for some two sets $$A$$ ,$$B$$ can we say something about the stationary condition for $$C=A\cap B$$?
i.e if we look at $$\mathbb{R}_n^+\cap B[0,r]$$ can we get the conditions in easier way than checking again by definition?