Just as curves have high and low points, surfaces have high and low points. Finding these extrema in 3D is similar to finding extrema in 2D, but, as with previoius topics, there are more options for extrema.
Subsection2.6.1Review
First, review the process of finding extrema in 2D. Why this works can be extended to higher dimensions.
Activity40.
We review what information derivatives tell us about curves, and specifically how relative extrema are identified.
Consider the function \(f(x)=3x^4-52x^3+282x^2-420x+925\text{.}\) We will look for relative extrema, so we do not use an interval.
(a)
Calculate \(f^\prime(x)\text{.}\)
(b)
Determine where extrema might occur using \(f^\prime(x)\text{.}\)
(c)
Determine on which intervals \(f(x)\) is increasing and decreasing.
(d)
Identify where relative maxima and minima occur.
(e)
How does the first derivative identify extrema?
Subsection2.6.2Experiment
Activity41.
Here we attempt to use the same idea to identify extrema on surfaces and discover a bit about what is different.
For the following questions use the surface \(f(x,y)=y^3-y+x^3-x\text{.}\)
(a)
Calculate \(\frac{\partial f}{\partial x} \) and \(\frac{\partial f}{\partial y}. \)
(b)
Determine where extrema might occur using \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) in a fashion similar to the mono-variate method.
(c)
Identify the relative maxima and minima by comparing your results above to a graph.
(d)
Can relative maxima and minima be identified using only the first partials?
Figure2.6.1.Illustration of Surface Relative Extrema
Subsection2.6.3Method
Definition2.6.2.Maximum.
A function \(f:\R^n \to \R\) has a maximum at \(A \in \R^n\) if and only if for some \(\delta,\)\(\|X-A\| \lt \delta \) implies \(f(A) \ge f(X).\)
Activity42.
(a)
Explain how this definition matches the definition for maximum on 2D functions. You can consider the directional derivative concept.
(b)
Write a definition for a minimum for a function \(f:\R^n \to \R\text{.}\)
Theorem2.6.3.Critical Values.
If a function \(f(x,y)\) has an extreme value at \((x,y)\) then \(f_x(x,y)=0\) and \(f_y(x,y)=0.\)
Theorem2.6.4.Extrema.
If the second partials are continuous about \((a,b),\) and \(f_x(a,b)=0,\)\(f_y(a,b)=0\) then for \(D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2\)
If \(D \gt 0\) and \(f_{xx}(a,b) \gt 0,\) a minimum occurs at \((a,b)\)
If \(D \gt 0\) and \(f_{xx}(a,b) \lt 0,\) a maximum occurs at \((a,b)\)
If \(D \lt 0\) a saddle point occurs at \((a,b).\)
Activity43.
(a)
How could \(D(a,b)\) be re-written under the conditions of Theorem 2.2.3?
(b)
Devise a mnemonic to remember \(D\) based on the result above.
ExercisesExercises
Use this theorem to solve the following.
1.
Identify extrema for \(f(x,y)=x^2+xy+y^2.\)
2.
Identify extrema for \(f(x,y)=x^3-x+y^3-y.\)
3.
Identify extrema for \(f(\alpha,\beta)=\sin \alpha + \sin \beta.\)
4.
Identify extrema for \(f(x,y)=\sin(x^2+y^2).\)
5.
Identify all high and low areas for \(f(x,y)=\sin(x^2+y^2).\)
