A function \(f(x, y)\) is said to have a relative maximum at the point \(P(a, b)\) in the domain of \(f\) if \(f(a, b) \geq f(x, y)\) for all points \((x, y)\) in a disk centered at \(P\). Similarly, if \(f(c, d) \leq f(x, y)\) for all points \((x, y)\) in a circular disk centered at \(Q\), then \(f(x, y)\) has a relative minimum at \(Q(c, d)\).
Critical Points
A point \((a,b)\) in the domain of \(f(x,y)\) is called a critical point of \(f\), if
\(f_x(a,b)=0 \quad \text{and} \quad f_y(a,b)=0,\)
or if one of these partial derivative does not exist.
Theorem: If \(f\) has a relative extreme value at \((a,b)\), and the partial derivatives \(f_x\) and \(f_y\) exist, then \(f_x(a,b)=0\) and \(f_y(a,b)=0\).
This theorem says that the relative extrema of \(f\) can occur only at critical points.
Saddle Points
Although all relative extrema of a function must occur at critical points, not every critical point yields a relative extremum. A critical point which is neither a local minimum nor a local maximum is called a saddle point. A saddle point a function indeed looks like a saddle: it has a relative maximum in some direction and a relative minimum in another direction (in simplest cases, such as \(f(x,y)=y^2-x^2\), these directions are the directions of the x-axis and the y-axis).
Finding Relative Extrema Using the Second Derivative Test
The procedure involving second-order derivatives can be used to decide whether a given critical point is a relative maximum, a relative minimum, or a saddle point.
Let \(f(x, y)\) be a function of two variables \(x\) and \(y\) whose partial derivatives \(f_x\), \(f_y\), \(f_{xx}\), \(f_{yy}\), and \(f_{xy}\) all exist and are continuous on a disk centred at each critical point, and let
\(D(x, y) = f_{xx}(x, y)f_{yy}(x, y) - [f_{xy}(x, y)]^2.\)
Step 1. Find all critical points of \(f(x, y)\), that is, all points \((a, b)\) so that
\(f_x(a, b) = 0 \quad \text{and} \quad f_y(a, b) = 0.\)
Step 2. For each critical point \((a, b)\) found in Step 1, evaluate \(D(a, b)\).
Step 3. If \(D(a, b) < 0\), then \((a, b)\) is a saddle point (and the function has no extreme value there).
If \(D(a, b) > 0\), compute \(f_{xx}(a, b)\):
If \(f_{xx}(a, b) > 0\), then \((a, b)\) is a relative minimum.
If \(f_{xx}(a, b) < 0\), then \((a, b)\) is a relative maximum.
If \(D(a, b) = 0\), the test is inconclusive and \(f\) may have either a relative extremum or a saddle point at \((a, b)\).
In summary:
| Sign of \(D\) | Sign of \(f_{xx}\) | Behavior at \((a,b)\) |
| + | + | relative minimum |
| + | - | relative maximum |
| - | not relevant | saddle point |
Example.
Find all critical points for the function \(f(x,y)=12x-x^3-4y^2\) and classify each as a relative maximum, a relative minimum, or a saddle point.
Solution.
Solving \(f_x=12-3x^2\) and \(f_y=-8y\) yields \(x=2\), \(x=-2\), and \(y=0\). Thus, the critical points are \((2,0)\) and \((-2,0)\). The second-order partial derivatives are \(f_{xx}=-6x\), and \(f_{xy}=0\).
Then, \(D(x,y)=f_{xx}f_{yy}-(f_{xy})^2=(-6x)(-8)-0=48x\).
At \((2,0): D(2,0)=48(2)=96>0\) and \(f_{xx}(2,0)=-6(2)=-12<0\) which means that \((2,0)\) is a relative maximum.
At \((-2,0): D(-2,0)=48(-2)=-96<0\) which means that \((-2,0)\) is a saddle point.
Example.
Find all critical points for the function \(f(x,y)=e^{2xy}\) and classify each as a relative maximum, a relative minimum, or a saddle point.
Solution.
Since \(f_x=2ye^{2xy}\) and \(f_y=2xe^{2xy}\), there is only one critical point \((0,0)\).
The second-order partial derivatives are \(f_{xx}=4y^2e^{2xy}\), \(f_{yy}=4x^2e^{2xy}\), and \(f_{xy}=2e^{2xy}+4xye^{2xy}\).
Then,
\(D(x,y)=f_{xx}f_{yy}-(f_{xy})^2\)
\(=(4y^2e^{2xy})(4x^2e^{2xy})-(2e^{2xy}+4xye^{2xy})^2\)
At \((0,0)\), \(D(0,0)=0-(2e^0+0)^2=-4<0\), which means that \((0,0)\) is a saddle point.