First-order necessary condition for a local minimizer

Question

Let C be a convex set in Rn, and let f be a differentiable function on an open set containing C . First-order necessary condition for a local minimizer : If x∗ ∈ C is a local minimizer of f on C, then the inner product <∇f(x∗), x − x∗> ≥ 0 for all x ∈ C. I cannot understand the last line of the proof , how does it take the inner product?

Answer 1

Consider $n=2$, $C=[0,1]\times[0,1]$, $f(x,y)=x+y$. Then $(0,0)$ is a local minmizer of $f$ on $C$, even though $\nabla f$ is non-zero