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6.10.E: Further Problems on Maxima and Minima

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Exercise $$\PageIndex{1}$$

Fill in all details in Examples 1 and 2 and the proofs of all theorems in this section.

Exercise $$\PageIndex{2}$$

Redo Example (B) in §9 by Lagrange's method.
[Hint: Set $$F(x, y, z)=f(x, y, z)-r\left(x^{2}+y^{2}+z^{2}\right), g(x, y, z)=x^{2}+y^{2}+z^{2}-1$$. Compare the values of $$f$$ at all critical points.]

Exercise $$\PageIndex{3}$$

An ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$
is cut by a plane $$u x+v y+w z=0.$$ Find the semiaxes of the section-ellipse, i.e., the extrema of
$\rho^{2}=[f(x, y, z)]^{2}=x^{2}+y^{2}+z^{2}$
under the constraints $$g=\left(g_{1}, g_{2}\right)=\overrightarrow{0},$$ where
$g_{1}(x, y, z)=u x+v y+w z \text { and } g_{2}(x, y, z)=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}-1.$
Assume that $$a>b>c>0$$ and that not all $$u, v, w=0$$.
[Outline: By Note 2, explore the rank of the matrix
$\left(\begin{array}{ccc}{x / a^{2}} & {y / b^{2}} & {z / c^{2}} \\ {u} & {v} & {z}\end{array}\right).$
(Why this particular matrix?)
Seeking a contradiction, suppose all its $$2 \times 2$$ determinants vanish at all points of the section-ellipse. Then the upper and lower entries in (14) are proportional (why?); so $$x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=0$$ (a contradiction!).
Next, set
$F(x, y, z)=x^{2}+y^{2}+z^{2}+r\left(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\right)+2 s(u x+v y+w z).$
Equate $$dF$$ to $$0:$$
$x+\frac{r x}{a^{2}}+s u=0, \quad y+\frac{r y}{b^{2}}+s v=0, \quad z+\frac{r z}{c^{2}}+s w=0.$
Multiplying by $$x, y, z,$$ respectively, adding, and combining with $$g=\overrightarrow{0},$$ obtain $$r=$$ $$-\rho^{2};$$ so, by (15), for $$a, b, c \neq \rho$$,
$x=\frac{-s u a^{2}}{a^{2}-\rho^{2}}, \quad y=\frac{-s v b^{2}}{b^{2}-\rho^{2}}, \quad z=\frac{-s w c^{2}}{c^{2}-\rho^{2}}.$
Find $$s, x, y, z,$$ then compare the $$\rho$$-values at critical points.]

Exercise $$\PageIndex{4}$$

Find the least and the largest values of the quadratic form
$f(\vec{x})=\sum_{i, k=1}^{n} a_{i k} x_{i} x_{k} \quad\left(a_{i k}=a_{k i}\right)$
on the condition that $$g(\vec{x})=|\vec{x}|^{2}-1=0\left(f, g : E^{n} \rightarrow E^{1}\right)$$.
[Outline: Let $$F(\vec{x})=f(\vec{x})-t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right).$$ Equating $$d F$$ to $$0,$$ obtain
$\begin{array}{l}{\left(a_{11}-t\right) x_{1}+a_{12} x_{2}+\ldots+a_{1 n} x_{n}=0,} \\ {a_{21} x_{1}+\left(a_{22}-t\right) x_{2}+\ldots+a_{2 n} x_{n}=0,} \\ {\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots} \\ {a_{n 1} x_{1}+a_{n 2} x_{2}+\ldots+\left(a_{n n}-t\right) x_{n}=0.}\end{array}$
Using Theorem 1(iv) in §6, derive the so-called characteristic equation of $$f$$,
$\left|\begin{array}{cccc}{a_{11}-t} & {a_{12}} & {\dots} & {a_{1 n}} \\ {a_{21}} & {a_{22}-t} & {\dots} & {a_{2 n}} \\ {\dots} & {\dots} & {\dots} & {\dots} \\ {a_{n 1}} & {a_{2 n}} & {\dots} & {a_{n n}-t}\end{array}\right|=0,$
of degree $$n$$ in $$t.$$ If $$t$$ is one of its $$n$$ roots (known to be real), then equations (16) admit a nonzero solution for $$\vec{x}=\left(x_{1}, \ldots, x_{n}\right);$$ by replacing $$\vec{x}$$ by $$\vec{x} /|\vec{x}|$$ if necessary, $$\vec{x}$$ satisfies also the constraint equation $$g(\vec{x})=|\vec{x}|^{2}-1=0.$$ (Explain!) Thus each root $$t$$ of (17) yields a critical point $$\vec{x}_{t}=\left(x_{1}, \ldots, x_{n}\right).$$
Now, to find $$f\left(\vec{x}_{t}\right),$$ multiply the $$k$$th equation in (16) by $$x_{k}, k=1, \ldots, n,$$ and add to get
$0=\sum_{i, k=1}^{n} a_{i k} x_{i} x_{k}-t \sum_{k=1}^{n} x_{k}^{2}=f\left(\vec{x}_{t}\right)-t.$
Hence $$f\left(\vec{x}_{t}\right)=t$$.
Thus the values of $$f$$ at the critical points $$\vec{x}_{t}$$ are simply the roots of (17). The largest (smallest) root is also the largest (least) value of $$f$$ on $$S=\left\{\vec{x} \in E^{n}| | \vec{x} |=1\right\}$$ (Explain!)]

Exercise $$\PageIndex{5}$$

Use the method of Problem 4 to find the semiaxes of
(i) the quadric curve in $$E^{2},$$ centered at $$\overrightarrow{0},$$ given by $$\sum_{i, k=1}^{2} a_{i k} x_{i} x_{k}=1;$$ and
(ii) the quadric surface $$\sum_{i, k=1}^{3} a_{i k} x_{i} x_{k}=1$$ in $$E^{3},$$ centered at (\overrightarrow{0}\).
Assume $$a_{i k}=a_{k I}$$.
[Hint: Explore the extrema of $$f(\vec{x})=|\vec{x}|^{2}$$ on the condition that
$g(\vec{x})=\sum_{i, k} a_{i k} x_{i} x_{k}-1=0.]$

Exercise $$\PageIndex{6}$$

Using Lagrange's method, redo Problems 4, 5, 6, 7, 11, 12, and 13 of §9.

Exercise $$\PageIndex{7}$$

In $$E^{2},$$ find the shortest distance from $$\overrightarrow{0}$$ to the parabola $$y^{2}=2(x+a)$$.

Exercise $$\PageIndex{8}$$

In $$E^{3}$$, find the shortest distance from $$\overrightarrow{0}$$ to the intersection line of two planes given by the formulas $$\vec{u} \cdot \vec{x}=a$$ and $$\vec{v} \cdot \vec{x}=b$$ with $$\vec{u}$$ and $$\vec{v}$$ different from $$\overrightarrow{0}.$$ (Rewrite all in coordinate form!)

Exercise $$\PageIndex{9}$$

In $$E^{n},$$ find the largest value of $$|\vec{a} \cdot \vec{x}|$$ if $$|\vec{x}|=1.$$ Use Lagrange's method.

Exercise $$\PageIndex{10*}$$

(Hadamard's theorem.) If $$A=\operatorname{det}\left(x_{i k}\right)(i, k \leq n),$$ then
$|A| \leq \prod_{i=1}^{n}\left|\vec{x}_{i}\right|,$
where $$\vec{x}_{i}=\left(x_{i 1}, x_{i 2}, \ldots, x_{i n}\right)$$.
[Hints: Set $$a_{i}=\left|\vec{x}_{i}\right|.$$ Treat $$A$$ as a function of $$n^{2}$$ variables. Using Lagrange's method, prove that, under the $$n$$ constraints $$\left|\vec{x}_{i}\right|^{2}-a_{i}^{2}=0, A$$ cannot have an extremum unless $$A^{2}=\operatorname{det}\left(y_{i k}\right),$$ with $$y_{I k}=0$$ (if $$i \neq k$$) and \(y_{ii}=a_{i}^{2}.]

6.10.E: Further Problems on Maxima and Minima is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.