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A critical point (a, b) of f(x, y) satisfies ∇f(a, b) = 0 (or ∇f is undefined). Critical points are candidates for local maxima, local minima, or saddle points. Every continuous function on a closed bounded set attains its absolute maximum and minimum.
From your prerequisite on the gradient, you know that ∇f at a point gives the direction and magnitude of steepest ascent. A critical point is where this steepest-ascent direction ceases to exist in the usual sense: the gradient is zero, meaning the function is instantaneously flat in every direction. In single-variable calculus you found critical points where f′(x) = 0; the multivariable condition ∇f = 0 is the exact generalization — it requires both ∂f/∂x = 0 and ∂f/∂y = 0 simultaneously.
The three types of critical point correspond to three distinct topographic shapes. A local minimum looks like the bottom of a bowl: the function rises in every direction away from the point. A local maximum looks like the top of a hill: the function falls in every direction. A saddle point looks like a mountain pass: the function rises in some directions and falls in others. The gradient is zero at all three, so the gradient condition alone cannot tell them apart — that requires additional information about the second-order behavior.
The second derivative test for two variables uses the Hessian matrix H, whose entries are the second partial derivatives: H = [[f_xx, f_xy], [f_yx, f_yy]]. The discriminant D = f_xx · f_yy − (f_xy)² captures the Hessian's determinant. If D > 0 and f_xx > 0, the point is a local minimum (bowl opening upward). If D > 0 and f_xx < 0, it's a local maximum (bowl opening downward). If D < 0, it's a saddle point. If D = 0, the test is inconclusive — higher-order methods are needed. The intuition: D > 0 means both principal curvatures have the same sign (pure bowl), while D < 0 means they have opposite signs (saddle).
For optimization on a closed bounded region, the story extends beyond interior critical points. By the extreme value theorem, a continuous function on a compact set attains its absolute extrema somewhere. The candidates are: all interior critical points where ∇f = 0, and all points on the boundary. The boundary is typically a curve, so you parameterize it and apply single-variable optimization there. Checking all candidates and comparing values gives the absolute maximum and minimum — this is the complete algorithm for constrained optimization on closed bounded domains.