Task 22.1

There are three optimization problems below. Each can be solved with a different method (first semester calculus, Lagrange multipliers, and the second derivative test with eigenvalues). Solve each problem below, and explain your choice of the method used.

  1. The elevation near a rover is given by $z=y+x^2$. The rover travels along a path given by $y-2x=5$. Find the $(x,y)$ location of any maxes or mins along the rover's path, and classify the point(s) appropriately.
  2. The elevation near a rover is given by $z=y+x^2$. The rover travels along the path parametrized by $\vec r(t) =(t,2t+5)$. Find the $(x,y)$ location of any maxes or mins along the rover's path, and classify the point(s) appropriately.
  3. The elevation near a rover is given by $f(x,y)=x^2+xy+y^2-2y$. Determine the location of any maxes or mins near the rover, and classify the point(s) appropriately.

Task 22.2

For each problem below, decide if you'll need to use Lagrange multipliers or the second derivative test. If you choose Lagrange multipliers, then state $f$, $g$, and $c$, along with the system of equations that must be solved. If you choose the second derivative test, then state $f$, $Df$, and $D^2f$. Then use the appropriate Mathematica notebook (either LagrangeMultipliers.nb or 2ndDerTest.nb) to solve the problem.

  1. Let $f(x,y)=x^3 + 3xy +y^3$. Find all local extreme values of $f$.
  2. Find the dimensions of the rectangle of largest possible area that will fit inside of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$.
  3. Find three numbers whose sum is 9 and whose sum of squares is minimized.
  4. Find the largest box in the first octant (all variables are positive) that can fit under the paraboloid $z=9-x^2-y^2$. The volume of such a box is given by $V=lwh = xyz = xy(9-x^2-y^2)$.
  5. A rover travels along a circle of radius 5, centered at the origin. The elevation of the surrounding hill is give by $z = 4x^2-4xy+y^2$. What are the highest and lowest elevations reached by the rover.

Task 22.3

In this task we'll derive the version of the second derivative test that is found in most multivariate calculus texts. The test given below only works for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$. The eigenvalue test you have been practicing will work with a function of the form $f:\mathbb{R}^n\to\mathbb{R}$, for any natural number $n$.

Suppose that $f(x,y)$ has a critical point at $(a,b)$.

  1. We know that $D^2f(a,b) = \begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}$, where all partials are evaluated at $(a,b)$. Prove that the eigenvalues of $D^2f(a,b)$ are given by $$\lambda = \frac{(f_{xx}+f_{yy})\pm \sqrt{(f_{xx}+f_{yy})^2 - 4(f_{xx}f_{yy}-f_{xy}^2)}}{2}.$$
  2. Let $D=f_{xx}f_{yy}-f_{xy}^2$.
    1. If $D<0$, explain why the eigenvalues differ in sign.
    2. If $D=0$, explain why zero is an eigenvalue.
    3. If $D>0$, explain why the eigenvalues must have the same sign.
    4. If $D>0$, and $f_{xx}>0$, explain why $f$ has a local minimum at $(a,b)$.
    5. If $D>0$, and $f_{xx}<0$, explain why $f$ has a local maximum at $(a,b)$.
    6. How would you interpret $f_{xx}$ in terms of concavity?
  3. The only critical point of $f(x,y) = x^2+3xy+2y^2$ is at $(0,0)$. Does this point correspond to a local maximum, local minimum, or saddle point? Find $D$ from part 2 to answer the question.

Task 22.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.