Homework 7

Due: March 6, 2018, 8 a.m.

Please give a complete, justified solution to each question below. A single-term answer without explanation will receive no credit.

Please complete each question on its own sheet of paper (or more if necessary), and upload to Gradsescope.

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1. Consider the function $$f(x, y) = 2 x^2+y^2 - k xy$$ where $k$ is some fixed constant.

  a. Show that for any value of $k$, $(0, 0)$ is a critical point of $f$.

  b. Determine the values of $k$ (if any) for which $(0, 0)$ is

  • a saddle point,
  • corresponds with a local maximum,
  • corresponds with a local minimum.

2. Show that the point $(c,d)$ for which the sum of the squares of the distances to a set of fixed points $(a_1,b_1),(a_2,b_2),\ldots,(a_n,b_n)$ is minimized is located such that $c$ is the average of the $a$-coordinates and $d$ is the average of the $b$-coordinates.

3. A region in the plane consist of a semicircle sitting atop a rectangle. If the perimeter is fixed and the area is maximized, find the ratio $\frac{\ell}{h}$ of the sides of the rectangle. HW7-2-Figure

4. A solid sphere $x^2+y^2+z^2\le1$ has a temperature distribution given by $$ T(x,y,z) = e^{-((2+5x)^2+(1-5y)^2+(2-5z)^2)} $$ Find its hottest and coldest temperatures.

5.  Consider the problem of minimizing the function $f(x, y) = x$ on the curve $y^2 + x^4 - x^3 = 0$ (a type of curve known as a "piriform"). Piriform

  a. Try using Lagrange multipliers to solve the problem.

  b. Show that the minimum value is $f(0, 0)$ even though the Lagrange condition $\nabla f(0, 0) = \lambda \nabla g(0, 0)$ is not satisfied for any value $\lambda$.

  c. Briefly explain (1–2 sentences) why Lagrange multipliers fail to find the minimum value in this case.