
# 6.9.E: Problems on Maxima and Minima


Exercise $$\PageIndex{1}$$

Verify Note 1.

Exercise $$\PageIndex{1'}$$

Complete the missing details in the proof of Theorems 2 and 3.

Exercise $$\PageIndex{2}$$

Verify Examples (A) and (B). Supplement Example (A) by applying Theorem 2.

Exercise $$\PageIndex{3}$$

Test $$f$$ for extrema in $$E^{2}$$ if $$f(x, y)$$ is
(i) $$\frac{x^{2}}{2 p}+\frac{y^{2}}{2 q}(p>0, q>0)$$;
(ii) $$\frac{x^{2}}{2 p}-\frac{y^{2}}{2 q}(p>0, q>0)$$;
(iii) $$y^{2}+x^{4}$$;
(iv) $$y^{2}+x^{3}$$.

Exercise $$\PageIndex{4}$$

(i) Find the maximum volume of an interval $$A \subset E^{3}$$ (see Chapter 3, §7) whose edge lengths $$x, y, z$$ have a prescribed sum: $$x+y+z=a$$.
(ii) Do the same in $$E^{4}$$ and in $$E^{n};$$ show that $$A$$ is a cube.
(iii) Hence deduce that
$\sqrt[n]{x_{1} x_{2} \cdots x_{n}} \leq \frac{1}{n} \sum_{1}^{n} x_{k} \quad\left(x_{k} \geq 0\right),$
i.e., the geometric mean of $$n$$ nonnegative numbers is $$\leq$$ their arithmetic mean.

Exercise $$\PageIndex{5}$$

Find the minimum value for the sum $$f(x, y, z, t)=x+y+z+t$$ of four positive numbers on the condition that $$x y z t=c^{4}$$ (constant).
[Answer: $$x=y=z=t=c; f_{\max }=4c$$.]

Exercise $$\PageIndex{6}$$

Among all triangles inscribed in a circle of radius $$R,$$ find the one of maximum area.
[Hint: Connect the vertices with the center. Let $$x, y, z$$ be the angles at the center. Show that the area of the triangle $$=\frac{1}{2} R^{2}(\sin x+\sin y+\sin z),$$ with $$z=2 \pi-(x+y)$$.]

Exercise $$\PageIndex{7}$$

Among all intervals $$A \subset E^{3}$$ inscribed in the ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$
find the one of largest volume.
[Answer: the edge lengths are $$\frac{2 a}{\sqrt{3}}, \frac{2 b}{\sqrt{3}}, \frac{2 c}{\sqrt{3}}$$.]

Exercise $$\PageIndex{8}$$

Let $$P_{i}=\left(a_{i} \cdot b_{i}\right), i=1,2,3,$$ be 3 points in $$E^{2}$$ forming a triangle in which one angle (say, $$\,easureangle P_{1})$$ is $$\geq 2 \pi / 3$$.
Find a point $$P=(x, y)$$ for which the sum of the distances,
$P P_{1}+P P_{2}+P P_{3}=\sum_{i=1}^{3} \sqrt{\left(x-a_{i}\right)^{2}+\left(y-b_{i}\right)^{2}},$
is the least possible.
[Outline: Let $$f(x, y)=\sum_{i=1}^{3} \sqrt{\left(x-a_{i}\right)^{2}+\left(y-b_{i}\right)^{2}}$$.
Show that $$f$$ has no partial derivatives at $$P_{1}, P_{2},$$ or $$P_{3}$$ (and so $$P_{1}, P_{2},$$ and $$P_{3}$$ are critical points at which an extremum may occur), while at other points $$P,$$ partials do exist but never vanish simultaneously, so that there are no other critical points.
Indeed, prove that $$D_{1} f(P)=0=D_{2} f(P)$$ would imply that
$\sum_{i=1}^{3} \cos \theta_{i}=0=\sum_{1}^{3} \sin \theta_{I},$
where $$\theta_{i}$$ is the angle between $$\overline{P P_{i}}$$ and the $$x$$-axis; hence
$\sin \left(\theta_{1}-\theta_{2}\right)=\sin \left(\theta_{2}-\theta_{3}\right)=\sin \left(\theta_{3}-\theta_{1}\right) \quad\text {(why?),}$
and so $$\theta_{1}-\theta_{2}=\theta_{2}-\theta_{3}=\theta_{3}-\theta_{1}=2 \pi / 3,$$ contrary to $$\angle P_{1} \geq 2 \pi / 3.$$ (Why?)
From geometric considerations, conclude that $$f$$ has an absolute minimum at $$P_{1}$$.
(This shows that one cannot disregard points at which $$f$$ has no partials.)]

Exercise $$\PageIndex{9}$$

Continuing Problem 8, show that if none of $$\angle P_{1}, \angle P_{2},$$ and $$\angle P_{3}$$ is $$\geq$$ $$2 \pi / 3,$$ then $$f$$ attains its least value at some $$P$$ (inside the triangle) such that $$\angle P_{1} P P_{2}=\angle P_{2} P P_{3}=\angle P_{3} P P_{1}=2 \pi / 3$$.
[Hint: Verify that $$D_{1} f=0=D_{2} f$$ at $$P$$.
Use the law of cosines to show that $$P_{1} P_{2}>P P_{2}+\frac{1}{2} P P_{1}$$ and $$P_{1} P_{3}>P P_{3}+\frac{1}{2} P P_{1}$$.
Adding, obtain $$P_{1} P_{3}+P_{1} P_{2}>P P_{1}+P P_{2}+P P_{3},$$ i.e., $$f\left(P_{1}\right)>f(P).$$ Similarly, $$f\left(P_{2}\right)>f(P)$$ and $$f\left(P_{3}\right)>f(P).$$
Combining with Problem 8, obtain the result.]

Exercise $$\PageIndex{10}$$

In a circle of radius $$R$$ inscribe a polygon with $$n+1$$ sides of maximum area.
[Outline: Let $$x_{1}, x_{2}, \ldots, x_{n+1}$$ be the central angles subtended by the sides of the polygon. Then its area $$A$$ is
$\frac{1}{2} R^{2} \sum_{k=1}^{n+1} \sin x_{k},$
with $$x_{n+1}=2 \pi-\sum_{k=1}^{n} x_{k}.$$ (Why?) Thus all reduces to maximizing
$f\left(x_{1}, \ldots, x_{n}\right)=\sum_{k=1}^{n} \sin x_{k}+\sin \left(2 \pi-\sum_{k=1}^{r_{k}} x_{k}\right),$
on the condition that $$0 \leq x_{k}$$ and $$\sum_{k=1}^{n} x_{k} \leq 2 \pi.$$ (Why?)
These inequalities define a bounded set $$D \subset E^{n}$$ (called a simplex). Equating all partials of $$f$$ to $$0,$$ show that the only critical point interior to $$D$$ is $$\vec{x}=\left(x_{1}, \ldots, x_{n}\right),$$ with $$x_{k}=\frac{2 \pi}{n+1}, k \leq n$$ (implying that $$x_{n+1}=\frac{2 \pi}{n+1},$$ too). For that $$\vec{x},$$ we get
$f(\vec{x})=(n+1) \sin [2 \pi /(n+1)].$
This value must be compared with the "boundary" values of $$f,$$ on the "faces" of the simplex D (see Note 4).
Do this by induction. For $$n=2,$$ Problem 6 shows that $$f(\vec{x})$$ is indeed the largest when all $$x_{k}$$ equal $$\frac{2 \pi}{n+1}.$$ Now let $$D_{n}$$ be the "face" of $$D,$$ where $$x_{n}=0.$$ On that face, treat $$f$$ as a function of only $$n-1$$ variables, $$x_{1}, \ldots, x_{n-1}$$.
By the inductive hypothesis, the largest value of $$f$$ on $$D_{n}$$ is $$n \sin (2 \pi / n).$$ Similarly for the other "faces." As $$n \sin (2 \pi / n)<(n+1) \sin 2 \pi /(n+1),$$ the induction is complete.
Thus, the area $$A$$ is the largest when the polygon is regular, for which
$\left.A=\frac{1}{2} R^{2}(n+1) \sin \frac{2 \pi}{n+1}.\right]$

Exercise $$\PageIndex{11}$$

Among all triangles of a prescribed perimeter $$2p,$$ find the one of maximum area.
[Hint: Maximize $$p(p-x)(p-y)(p-z)$$ on the condition that $$x+y+z=2 p$$.]

Exercise $$\PageIndex{12}$$

Among all triangles of area $$A,$$ find the one of smallest perimeter.

Exercise $$\PageIndex{13}$$

Find the shortest distance from a given point $$\vec{p} \in E^{n}$$ to a given plane $$\vec{u} \cdot \vec{x}=c$$ (Chapter 3, §§4-6). Answer:
$\pm \frac{\vec{u} \cdot \vec{p}-c}{|\vec{u}|}.$
[Hint: First do it in $$E^{3},$$ writing $$(x, y, z)$$ for $$\vec{x}$$.]