I was following this lecture on KKT conditions and on the 14:05 mark he denotes the $x^*$ as the optimal solution to the problem with the curve shown in green. The feasibility region is covered in shades and only the point $x^*$ is the one that maximizes $f$.
He then proceeds to draw the gradient vector at the point $x^*$ to denote the direction of steepest ascent and makes the gradient vector equal to the linear combination of the constraint gradients using Lagrange multipliers. Now, as far as I know, the gradient vector at any local optima is zero (correct me if I am wrong). Is there something I am missing, or is it a perspective issue?