Since sharing the same second partials means the two surfaces will share the same concavity (or curvature) at the critical point, this causes these quadratic approximation surfaces to share the same behavior as the function \(z = f(x, y)\) that they approximate at the point of tangency. To see why this will help us, consider that the quadratic approximation of a function of two variables (its 2nd-degree Taylor polynomial) shares the same first and second partials as the function it approximates at the chosen point of tangency (or center point). In order to develop a general method for classifying the behavior of a function of two variables at its critical points, we need to begin by classifying the behavior of quadratic polynomial functions of two variables at their critical points. The same is true for functions of more than one variable, as stated in the following theorem. In Calculus 1, we showed that extrema of functions of one variable occur at critical points. It attains its minimum value at the boundary of its domain, which is the circle \(x^2+y^2=16.\)
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