Local maximum and Local minimum
A function of two variables has a local maximum at \((a,b)\) if \(f(x,y) \leq f(a,b)\) when \((x,y)\) is near \((a,b)\). This means \(f(x,y) \leq f(a,b)\) for all points \((x,y)\) in some disk with center \((a,b)\). The number \(f(a,b)\) is called a local maximum value. If \(f(x,y) \geq f(a,b)\) when \((x,y)\) is near \((a,b)\), then \(f\) has a local minimum at \((a,b)\) and \(f(a,b)\) is a local minimum value.
If the inequalities hold for all points \((x,y)\) in the domain of \(f\), then \(f\) has an absolute maximum or absolute minimum at \((a,b)\)
Local max/min and first-order partial derivative
If \(f\) has a local maximum or minimum at \((a,b)\) and the first-order partial derivatives of \(f\) exist there, then \(f_x(a,b)=0\) and \(f_y(a,b)=0\). We call the point \((a,b)\) a critical point.
Second Derivatives Test
Suppose the second partial derivatives of \(f\) are continuous on a disk with center \((a,b)\), and suppose that \(f_x(a,b)=0\) and \(f_y(a,b)=0\) [that is, \((a,b)\) is a critical point of \(f\)]. Let
\[D = D(a,b) = f*{xx}(a,b)f*{yy}(a,b) - f\_{xy}(a,b)^2\] \[\text{or}\] \[D = \begin{vmatrix} f*{xx} & f*{xy} \\\ f*{yx} & f*{yy} \end{vmatrix} = f*{xx}f*{yy} - f\_{xy}^2\](i) If \(D > 0\) and \(f_{xx}(a,b) > 0\), then \(f(a,b)\) is a local minimum
(ii) If \(D > 0\) and \(f_{xx}(a,b) < 0\), then \(f(a,b)\) is a local maximum
(iii) If \(D < 0\), then \(f(a,b)\) is not a local maximum or minimum.
If \(D=0\), the test gives no information: \(f\) could have a local maximum or minimum or could be a saddle point at \((a,b)\)
Saddle Point
In case (iii), the point \((a,b)\) is called a saddle point of \(f\).
Absolute Maximum an Minimum Values
Extreme Value Theorem
If \(f\) is continuous on a closed, bounded set \(D\) in \(\mathbb{R}^2\), then \(f\) attains an absolute maximum value \(f(x_1,y_1)\) and an absolute minimum value \(f(x_2,y_2)\) at some points \((x_1, y_1)\) and \((x_2, y_2)\) in \(D\).
To find the absolute maximum and minimum values of a continuous function \(f\) on a closed, bounded set \(D\):
A closed set in \(\mathbb{R}^2\) is one that contains all its boundary points. A boundary point of \(D\) is a point \((a,b)\) such that every disk with center \((a,b)\) contains points in \(D\) and also points not in \(D\).
A bounded set in \(\mathbb{R}^2\) is one that is contained within some disk.
- Find the values of \(f\) at the critical points of \(f\) in \(D\).
- Find the extreme values of \(f\) on the boundary of \(D\)
- The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value
References
[1] Stewart Calculus, 8th edition