As the first example, consider the function \(f(x,y) = x^2 + y^2\text{.}\) We have already studied this function and we know that it represents a paraboloid. Note that we can find a disk centered at the point \((0,0)\) such that \(f(0,0) \le f(x,y)\) for all the points \((x,y)\) inside the disk, as shown in Figure 8.3, and hence by definition \(f(0,0)\) is a local minimum of \(f(x,y) = x^2 + y^2\text{,}\) as shown in Figure 8.2.
Figure8.2.The paraboloid \(z = x^2 + y^2\text{.}\) Over an open disk centered at \((0,0)\text{,}\) the value \(f(0,0) = 0\) is smaller than every other value of the function, so \(f(0,0)\) is a local minimum.
Figure8.3.An open disk in the domain of \(f(x,y) = x^2 + y^2\) centered at \((0,0)\text{.}\) For every point \((x,y)\) inside the disk we have \(f(0,0) \le f(x,y)\text{,}\) so \(f(0,0)\) is a local minimum.
The next example is the function \(f(x,y) = -x^2 - y^2\text{.}\) Note that we can find a disk centered at the point \((0,0)\) such that \(f(0,0) \ge f(x,y)\) for all the points \((x,y)\) inside the disk and hence by definition \(f(0,0)\) is a local maximum of \(f(x,y) = -x^2 - y^2\text{,}\) as shown in Figure 8.4 and Figure 8.5.
Figure8.4.The paraboloid \(z = -x^2 - y^2\) opens downward. Over an open disk centered at \((0,0)\text{,}\) the value \(f(0,0) = 0\) is larger than every other value of the function, so \(f(0,0)\) is a local maximum.
Figure8.5.The trace of the surface \(z = -x^2 - y^2\) in the plane \(y = 0\) is the parabola \(z = -x^2\text{,}\) which has a maximum at the origin. By symmetry, every vertical cross-section through the origin has the same shape.
As an example, consider the function \(f(x,y) = x^2 + y^2\text{,}\) which has a local minimum at \((0,0)\text{.}\) You can easily check that \(f_x(0,0) = f_y(0,0) = 0\text{.}\)
An interior point of the domain of a function \(f(x,y)\) where both \(f_x\) and \(f_y\) are zero or where one or both of \(f_x\) and \(f_y\) do not exist is a critical point of \(f\text{.}\) Note that not every critical point is a local extremum.
A differentiable function \(f(x,y)\) has a saddle point at a critical point \((a,b)\) if in every open disk centered at \((a,b)\) there are domain points \((x,y)\) where \(f(a,b) \lt f(x,y)\) and domain points \((x,y)\) where \(f(a,b) \gt f(x,y)\text{.}\) The corresponding point \((a,b,f(a,b))\) on the surface \(z = f(x,y)\) is called a saddle point of the surface.
