Inequalities via constrained extrema

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Problem 1419 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.1 "The Theorems of Rolle, Lagrange, and Cauchy. Increase and Decrease of Functions. Inequalities"

In the following problems (1419--1420) it is useful to apply a change of coordinates.

Prove that on the circle $x^{2}+y^{2}=1$ the inequality

\frac{1}{a^{2}} \leq Ax^{2}+Bxy+Cy^{2} \leq \frac{1}{b^{2}},

holds, if the curve $Ax^{2}+Bxy+Cy^{2}=1$ is an ellipse with semi-axes $a$ and $b$ .

Problem 1420 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.1 "The Theorems of Rolle, Lagrange, and Cauchy. Increase and Decrease of Functions. Inequalities"

The surface $Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz=1$ , after a suitable rotation of axes, has the equation $A_{1}x_{1}^{2}+B_{1}y_{1}^{2}+C_{1}z_{1}^{2}=1$ , where $A_{1} < B_{1} < C_{1}$ . Prove the inequality

A_{1} \leq \frac{Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz}{x^{2}+y^{2}+z^{2}} \leq C_{1}.

Problem 1830 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.9 "Extremal Values of Functions of Several Variables"

Hadamard's inequality for the determinant of the third order

u = \begin{vmatrix} a & b & c \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{vmatrix}

has the form

\left|u\right| \leqslant 1, \quad \text{if } a^{2}+b^{2}+c^{2} = 1,\ a_{1}^{2}+b_{1}^{2}+c_{1}^{2} = 1,\ a_{2}^{2}+b_{2}^{2}+c_{2}^{2} = 1.

Prove this inequality.

Problem 1831 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.9 "Extremal Values of Functions of Several Variables"

Maclaurin's inequalities: if $x_{1}, x_{2}, \ldots , x_{n} \geqslant 0$ and $x_{1}+x_{2}+\ldots +x_{n} = na$ , then

x_{1}x_{2}+x_{1}x_{3}+\ldots +x_{n-1}x_{n} \leqslant \frac{n(n-1)}{1\cdot 2}a^{2},

x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+\ldots +x_{n-2}x_{n-1}x_{n} \leqslant \frac{n(n-1)(n-2)}{1\cdot 2\cdot 3}a^{3},

and so on. Prove these inequalities.

Problem 1832 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.9 "Extremal Values of Functions of Several Variables"

Huygens's problem (see problem 1807) can be reduced to proving the inequality:

(1+u_{1})(1+u_{2})\ldots (1+u_{n}) \geqslant (1+q)^{n}; \quad u_{k} > 0,

if $u_{1}u_{2}\ldots u_{n} = q_{n}$ . Prove this inequality.

Problem 1836 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.9 "Extremal Values of Functions of Several Variables"

Prove that in a triangle the radius of the inscribed circle is no greater than half the radius of the circumscribed circle.

References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.