3.5.E: Extreme Values (Exercises)

Exercise \(\PageIndex{1}\)

Find the maximum and minimum values of \(f(x, y)=x y\) on the set \(D=\{(x, y): \left.x^{2}+y^{2} \leq 1\right\} . \)

Answer

Maximum value of \(\frac{1}{2}\) at \(\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) and \(\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)\);

minimum value of \(-\frac{1}{2}\) at \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\) and \(\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)\)

Exercise \(\PageIndex{2}\)

Find the maximum and minimum values of \(f(x, y)=8-x^{2}-y^{2}\) on the set \(D = \left\{(x, y): x^{2}+9 y^{2} \leq 9\right\} . \)

Exercise \(\PageIndex{3}\)

Find the maximum and minimum values of \(f(x, y)=x^{2}+3 x y+y^{2}\) on the set \(D= \left\{(x, y): x^{2}+y^{2} \leq 4\right\} \).

Answer

Maximum value of 10 at \((\sqrt{2}, \sqrt{2})\) and \((-\sqrt{2},-\sqrt{2})\);

minimum value of -2 at \((-\sqrt{2}, \sqrt{2})\) and \((\sqrt{2},-\sqrt{2})\)

Exercise \(\PageIndex{4}\)

Find all local extreme values of \(f(x, y)=x e^{-\left(x^{2}+y^{2}\right)}\).

Exercise \(\PageIndex{5}\)

Find all local extreme values of \(g(x, y)=x^{2} e^{-\left(x^{2}+y^{2}\right)}\).

Answer

Local minimum of 0 at all points of the form \((0,y)\), \(-\infty<y<\infty\); local maximum of \(e^{-1}\) at (1,0) and (-1,0)

Exercise \(\PageIndex{6}\)

Find all local extreme values of \(g(x, y)=\frac{1}{1+x^{2}+y^{2}}\).

Exercise \(\PageIndex{7}\)

Find all local extreme values of \(f(x, y)=4 x y-2 x^{2}-y^{4}\).

Answer

Local maximum of 1 at (1,1) and (−1,−1); saddle point at (0,0)

Exercise \(\PageIndex{8}\)

Find all local extreme values of \(h(x, y)=2 x^{4}+y^{4}-x^{2}-2 y^{2} .\)

Exercise \(\PageIndex{9}\)

Find all local extreme values of \(f(x, y, z)=x^{2}+y^{2}+z^{2}\).

Answer

Local minimum of 0 at (0,0,0)

Exercise \(\PageIndex{10}\)

Find all local extreme values of \(g(x, y, z)=x^{2}+y^{2}-z^{2} .\)

Exercise \(\PageIndex{11}\)

A farmer wishes to build a rectangular bin, with a top, to hold a volume of 1000 cubic meters. Find the dimensions of the bin that will minimize the amount of material needed in its construction.

Answer

10 meters \(\times\) 10 meters \(\times\) 10 meters

Exercise \(\PageIndex{12}\)

A farmer wishes to build a rectangular bin, with a top, using 600 square meters of material. Find the dimensions of the bin that will maximize the volume.

Answer

8.43 meters \(\times\) 8.43 meters \(\times\) 8.43 meters

Exercise \(\PageIndex{13}\)

Find the extreme values of \(f(x, y, z)=x+y+z\) on the sphere with equation \(x^{2}+y^{2}+z^2 = 1\)

Answer

Maximum value of \(\sqrt{3}\) at \(\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\); minimum value of \(-\sqrt{3}\) at \(\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)\)

Exercise \(\PageIndex{14}\)

Find the minimum distance in \(\mathbb{R}^2\) from the origin to the line with equation \(3 x+2 y=4\).

Exercise \(\PageIndex{15}\)

Find the minimum distance in \(\mathbb{R}^3\) from the origin to the plane with equation \(2x+4y+z=6\).

Answer

Minimum distance of \(\frac{2 \sqrt{21}}{7}\) at \(\frac{2}{7}(2,4,1)\)

Exercise \(\PageIndex{16}\)

Find the minimum distance in \(\mathbb{R}^2\) from the origin to the curve with equation \(xy=1\).

Exercise \(\PageIndex{17}\)

The ellipsoid with equation \(x^{2}+2 y^{2}+z^{2}=4\) is heated so that its temperature at \((x,y,z)\) is given by \(T(x, y, z)=70+10(x-z)\). Find the hottest and coldest points on the ellipsoid.

Answer

Hottest point: \(98.28^{\circ}\) at \((\sqrt{2}, 0,-\sqrt{2})\); coldest point: \(41.72^{\circ}\) at \((-\sqrt{2}, 0, \sqrt{2})\)

Exercise \(\PageIndex{18}\)

Suppose an airline requires that the sum of the length, width, and height of carry-on luggage cannot exceed 45 inches (assuming the luggage is in the shape of a rectangular box). Find the dimensions of a piece of carry-on luggage that has the maximum volume.

Exercise \(\PageIndex{19}\)

Let \(f(x, y)=\left(y-4 x^{2}\right)\left(y-x^{2}\right)\).

(a) Verify that (0,0) is a critical point of \(f\).

(b) Show that \(Hf(0,0)\) is nondefinite.

(c) Show that along any line through the origin, \(f\) has a local minimum at (0,0).

(d) Find a curve through the origin such that, along the curve, \(f\) has a local maximum at (0,0). Note that this shows that (0,0) is a saddle point.

Exercise \(\PageIndex{20}\)

Let \(f(x, y)=(x-y)^{2}\). Find all critical points of \(f\) and categorize them according as they are either saddle points or the location of local extreme values. Is the second derivative test useful in this case?

Answer

Local minimum of 0 at all points of the form \((x,x)\), \(-\infty<x<\infty\)

Exercise \(\PageIndex{21}\)

Let \(g(x, y)=\sin \left(x^{2}+y^{2}\right)\). Find all critical points of \(g\). Which critical points are the location of local maximums? Local minimums? Are there any saddle points?

Exercise \(\PageIndex{22}\)

What does a plot of the gradient vectors look like around a saddle point of a function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R} ?\) You might look at some examples, like \(f(x, y)=x^{2}-y^{2}, f(x, y)=x y\), or even \(f(x, y)=x y e^{-\left(x^{2}+y^{2}\right)}\).

Exercise \(\PageIndex{23}\)

Given \(n\) points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) in \(\mathbb{R}^2\), the line with equation \(y=mx+b\) which minimizes

\[ L(m, b)=\sum_{i=1}^{n}\left(y_{1}-\left(m x_{i}+b\right)\right)^{2} \nonumber \]

is called the least squares line.

(a) Give a geometric interpretation for \(L(m,b)\).

(b) Show that the parameters of the least squares line are

\[ m=\frac{n \sum_{i=1}^{n} x_{i} y_{i}-\left(\sum_{i=1}^{n} x_{i}\right)\left(\sum_{i=1}^{n} y_{i}\right)}{n \sum_{i=1}^{n} x_{i}^{2}-\left(\sum_{i=1}^{n} x_{i}\right)^{2}} \nonumber \]

and

\[ b=\bar{y}-m \bar{x} , \nonumber \]

where

\[ \bar{y}=\frac{1}{n} \sum_{i=1}^{n} y_{i} \nonumber \]

and

\[ \bar{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i} . \nonumber \]

Exercise \(\PageIndex{24}\)

The following table is taken from a report prepared in the 1960’s to study the effect of leaks of radioactive waste from storage bins at the nuclear facilities at Hanford, Washington, on the cancer rates in nine Oregon counties which border the Columbia River. The table gives an index of exposure, which takes into account such things as distance from the Hanford facilities and the distance of the population from the river, along with the cancer mortality rate per 100,000 people.

County	Index of Exposure	Cancer Mortality Rate
Umatilla	2.49	147.1
Morrow	2.57	130.1
Gilliam	3.41	129.9
Sherman	1.25	113.5
Wasco	1.62	137.5
Hood River	3.83	162.3
Portland	11.64	207.5
Columbia	6.41	177.9
Clatsop	8.34	210.3

Using Exercise 22, find the least squares line for this data (let the index of exposure be the \(x\) data). Plot the points along with the line.

Answer

\(y=9.23 x+114.72\)