Why should the gradient be $0$ at a critical point? I'm not sure the reason other then we set the gradient of a function to zero to find the critical points.
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$\begingroup$The critical point of $f(x,y)$ is defined where $f_x = 0 = f_y$,and the gradient of $f$ is defined $\nabla f = (f_x,f_y)$. Thus it would be zero at critical point. It should be a comment but I tried capitalize on it...
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