Math Problems Vault

Problem 2.2.11 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.2 "Differential Calculus"

(1)(2)

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ be a function whose partial derivatives of order $\leq 2$ are everywhere defined and continuous.

?

(1)

Let $a \in \mathbb {R}^{n}$ be a critical point of $f$ (i.e., $\dfrac {\partial f}{\partial x_j}(a) = 0$ , $i = 1, \ldots , n$ ). Prove that $a$ is a local minimum provided the Hessian matrix

\left(\frac{\partial ^2 f}{\partial x_i \partial x_j}\right)

is positive definite at $x = a$ .

(2)

Assume the Hessian matrix is positive definite at all $x$ . Prove that $f$ has, at most, one critical point.

Problem 2.2.12 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.2 "Differential Calculus"

Prove that a real valued $C^3$ function $f$ on $\mathbb {R}^{2}$ whose Laplacian,

\frac{\partial ^2 f}{\partial x^2} + \frac{\partial ^2 f}{\partial y^2},

is everywhere positive cannot have a local maximum.

?

Problem 2.2.32 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 2.2 "Differential Calculus"

Let $f : \mathbb {R}^{3} \rightarrow \mathbb {R}$ be such that

f^{-1}(0) = \left\{ v \in \mathbb {R}^{3} \mid \left\| v\right\| = 1\right\} .

Suppose $f$ has continuous partial derivatives of orders $\leq 2$ . Is there a $p \in \mathbb {R}^{3}$ with $\left\| p\right\| \leq 1$ such that

\frac{\partial ^2 f}{\partial x^2}(p) + \frac{\partial ^2 f}{\partial y^2}(p) + \frac{\partial ^2 f}{\partial z^2}(p) \geq 0 \, ?

?

Problem 3621 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{2}+(y-1)^{2}$ .

?

Problem 3622 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{2}-(y-1)^{2}$ .

?

Problem 3623 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=(x-y+1)^{2}$ .

?

Problem 3624 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{2}-xy+y^{2}-2x+y$ .

?

Problem 3625 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{2}y^{3}(6-x-y)$ .

?

Problem 3626 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{3}+y^{3}-3xy$ .

?

Problem 3627 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{4}+y^{4}-x^{2}-2xy-y^{2}$ .

?

Problem 3628 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=xy+\dfrac {50}{x}+\dfrac {20}{y}$ $(x>0, y>0)$ .

?

Problem 3629 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=xy\sqrt{1-\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}}$ $(a>0, b>0)$ .

?

Problem 3630 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=\dfrac {ax+by+c}{\sqrt{x^{2}+y^{2}+1}}$ $(a^{2}+b^{2}+c^{2} \neq 0)$ .

?

Problem 3631 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=1-\sqrt{x^{2}+y^{2}}$ .

?

Problem 3632 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=e^{2x+3y}(8x^{2}-6xy+3y^{2})$ .

?

Problem 3633 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=e^{x^{2}-y}(5-2x+y)$ .

?

Problem 3634 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=(5x+7y-25)e^{-(x^{2}+xy+y^{2})}$ .

?

Problem 3635 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x^{2}+xy+y^{2}-4\ln x-10\ln y$ .

?

Problem 3636 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=\sin x+\cos y+\cos (x-y)$ $\left(0 \leqslant x \leqslant \dfrac {\pi }{2}; 0 \leqslant y \leqslant \dfrac {\pi }{2}\right)$ .

?

Problem 3637 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=\sin x \sin y \sin (x+y)$ $(0 \leqslant x \leqslant \pi ; 0 \leqslant y \leqslant \pi )$ .

?

Problem 3638 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x-2y+\ln \sqrt{x^{2}+y^{2}}+3\operatorname {arctg}\dfrac {y}{x}$ .

?

Problem 3639 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=xy\ln (x^{2}+y^{2})$ .

?

Problem 3640 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=x+y+4\sin x\sin y$ .

?

Problem 3641 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $z=(x^{2}+y^{2})e^{-(x^{2}+y^{2})}$ .

?

Problem 3642 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $u=x^{2}+y^{2}+z^{2}+2x+4y-6z$ .

?

Problem 3643 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.7 "Extrema of Functions of Several Variables"

Investigate the extrema of the following function of several variables: $u=x^{3}+y^{2}+z^{2}+12xy+2z$ .

?

Problem 3627.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 3627 "[NO TITLE]"

[NO TASK]

Problem 3.5.1 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = x^{2}+x y+y^{2}-12 x-3 y$ ;

(2)

$u = 3+2 x-y-x^{2}+x y-y^{2}$ ;

(3)

$u = 3 x+6 y-x^{2}-x y+y^{2}$ ;

(4)

$u = 4 x^{2}-4 x y+y^{2}+4 x-2 y+1$ .

Problem 3.5.2 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = 3\left(x^{2}+y^{2}\right)-x^{3}+4 y$ ;

(2)

$u = 3 x^{2} y+y^{3}-12 x-15 y+3$ ;

(3)

$u = 2 x^{3}+x y^{2}+5 x^{2}+y^{2}$ ;

(4)

$u = 3 x^{3}+y^{3}-3 y^{2}-x-1$ ;

(5)

$u = x^{3}+y^{3}+3 a x y$

Problem 3.5.3 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = x^{2} y^{2}-2 x y^{2}-6 x^{2} y+12 x y$ ;

(2)

$u = x^{4}+y^{4}-2 x^{2}$ ;

(3)

$u = x^{4}+y^{4}-2(x-y)^{2}$ ;

(4)

$u = 2 x^{4}+y^{4}-x^{2}-2 y^{2}$ ;

(5)

$u = x y^{2}(12-x-y), x > 0, y > 0$ ;

(6)

$u = x^{2} y^{3}(6-x-y)$

Problem 3.5.4 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = (x+y) /(x y)-x y$ ;

(2)

$u = 8 / x+x / y+y$ ;

(3)

$u = 81(1 / x+1 / y)-\left(x^{2}+x y+y^{2}\right)$ ;

(4)

$u = x y+a / x+b / y$ .

Problem 3.5.5 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = 3 x^{2}-2 x \sqrt{y}+y-8 x$ ;

(2)

$u = x \sqrt{1+y}+y \sqrt{1+x}, x > -1, y > -1$ ;

(3)

$u = 1+x^{2}+\sqrt[3]{(y+2)^{2}}$ ;

(4)

$u = 1+y^{2}-\sqrt[5]{(x-2)^{4}}$

(5)

$u = x y \sqrt{12-4 x^{2}-y^{2}}$

(6)

$u = \frac{a x+y b+c}{\sqrt{1+x^{2}+y^{2}}}, a^{2}+b^{2}+c^{2} > 0$ .

Problem 3.5.6 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = \left(x+y^{2}\right) e^{x / 2}$ ;

(2)

$u = \left(x^{2}-2 y^{2}\right) e^{x-y}$

(3)

$u = \left(8 x^{2}-6 x y+3 y^{2}\right) e^{2 x+3 y}$ ;

(4)

$u = (5-2 x+y) e^{x^{2}-y}$

(5)

$u = x^{3} / 3+3 x^{2} e^{y}-e^{-y^{2}}$ ;

(6)

$u = (25-5 x-7 y) e^{-\left(x^{2}+x y+y^{2}\right)}$ ;

(7)

$u = \left(a x^{2}+b y^{2}\right) e^{-\left(x^{2}+y^{2}\right)}, a > 0, b > 0$ .

Problem 3.5.7 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = x^{2}+x y+y^{2}-4 \ln x-10 \ln y$ ;

(2)

$u = 108 \ln x-x y^{2}+y^{3} / 3$ ;

(3)

$u = x^{2}+y^{2}-32 \ln (x y)$ ;

(4)

$u = x y \ln \left(x^{2}+y^{2}\right)$ .

Problem 3.5.8 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate the function $u(x ; y)$ for extrema.

?

(1)

$u = \sin x+\cos y+\cos (x-y), x \in (0 ; \pi / 2), y \in (0 ; \pi / 2)$ ;

(2)

$u = \sin x \sin y \sin (x+y), x \in (0 ; \pi ), y \in (0 ; \pi )$ ;

(3)

$u = x+y+4 \sin x \sin y$ ;

(4)

$u = \left(1+e^{y}\right) \cos x-y e^{y}$ .

Problem 3.5.9 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Find all stationary points of the function $u = x^{4}+y^{4}-2 x^{2}$ and investigate it for extrema. Can the sufficient conditions for a strict extremum be used here?

?

Problem 3.5.10 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)

Prove that the function $u = \left(y^{2}-x\right)\left(y^{2}-2 x\right)$ :

?

(1)

along every line passing through the point $(0 ; 0)$ , has a minimum at this point;

(2)

has no minimum at the point $(0 ; 0)$ .

Problem 3.5.11 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Can a continuously differentiable function $u(x ; y)$ have infinitely many strict maxima and no minimum?

?

Problem 3.5.12 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Is the following statement true: if a continuously differentiable function $u(x ; y),(x ; y) \in R^{2}$ , has only one stationary point $\left(x_{0} ; y_{0}\right)$ , at which it has a local minimum, then the inequality $u(x ; y) \geqslant u\left(x_{0} ; y_{0}\right),(x ; y) \in R^{2}$ , holds?

?

Problem 3.5.13 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate the function $u(x ; y ; z)$ for extrema.

?

(1)

$u = x^{2}+y^{2}+(z+1)^{2}-x y+x$ ;

(2)

$u = 8-6 x+4 y-2 z-x^{2}-y^{2}-z^{2}$ ;

(3)

$u = x^{2}+y^{2}-z^{2}-4 x+6 y-2 z$ ;

(4)

$u = x^{3}+y^{2}+z^{2}+6 x y-4 z$ ;

(5)

$u = z y z(16-x-y-2 z)$

(6)

$u = x y^{2} z^{3}(49-x-2 y-3 z)$ .

Problem 3.5.14 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate the function $u(x ; y ; z)$ for extrema.

?

(1)

$u = \frac{x y+x z^{2}+y^{2} z}{x y z}+x+1$ ;

(2)

$u = \frac{256}{x}+\frac{x^{2}}{y}+\frac{y^{2}}{z}+z^{2}$

(3)

$u = x+\frac{y^{2}}{4 x}+\frac{z^{2}}{y}+\frac{2}{z}$

(4)

$u = \frac{x^{2}}{y z}+\frac{y^{2}}{x z}+\frac{z^{2}}{x y}$ .

Problem 3.5.15 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)

Investigate the function $u(x ; y ; z)$ for extrema.

?

(1)

$u = \sin x+\sin y+\sin z-\sin (x+y+z), x, y, z \in (0 ; \pi )$ ;

(2)

$u = (x+7 z) e^{-\left(x^{2}+y^{2}+z^{2}\right)}$ ;

(3)

$u = 2 \ln x+3 \ln y+5 \ln z+\ln (22-x-y-z)$ .

Problem 3.5.16 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)

Investigate the function $u(x), x \in R^{n}, x_{k} > 0, k = 1,2, \ldots , n$ , for extrema:

?

(1)

$u = x_{1} x_{2}^{2} \ldots x_{n}^{n}\left(1-\sum_{k = 1}^{n} k x_{k}\right)$ ;

(2)

$u = \sum_{k = 0}^{n} \frac{x_{k+1}}{x_{k}}, x_{0} = a > 0, x_{n+1} = b > 0$ .

Problem 3.5.17 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

Investigate for extrema the continuously differentiable function $u = u(x ; y)$ defined implicitly by the conditions:

?

(1)

$x^{2}+y^{2}+u^{2}-4 x-6 y-4 u+8 = 0, u > 2$ ;

(2)

$25 x^{2}+y^{2}+16 u^{2}-50 x+64 u-311 = 0, u < -2$ ;

(3)

$x^{2}+4 y^{2}+9 u^{2}-6 x+8 y-36 u = 0, u > 2$ ;

(4)

$\left(x^{2}+y^{2}+u^{2}\right)^{2} = 8\left(x^{2}+y^{2}-u^{2}\right), u > 0$

(5)

$\left(x^{2}+y^{2}+u^{2}+9\right)^{2} = 100\left(x^{2}+y^{2}\right), u < 0$ .

Problem 3.5.18 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.5.2 "Problems"

(1)(2)(3)(4)

Investigate for a strict extremum each continuously differentiable function $u = u(x ; y)$ defined implicitly by the equation:

?

(1)

$x^{2}+y^{2}+u^{2}+2 x-2 y+4 u-3 = 0$ ;

(2)

$2 x^{2}+2 y^{2}+u^{2}+8 y u-u+8 = 0$ ;

(3)

$x^{3}-y^{2}+u^{2}-3 x+4 y+u-8 = 0$ ;

(4)

$\left(x^{2}+y^{2}\right)^{2}+u^{4}-8\left(x^{2}+y^{2}\right)-10 u^{2}+16 = 0$ .

Problem 1.9.8 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.9 "Functions of Several Variables"

Suppose that $f$ is a diffierentiable real function in an open set $E \subset \mathbb {R}^{n}$ , and that $f$ has a local maximum at a point $\mathbf{x} \in E$ . Prove that $f^{\prime }(\mathbf{x}) = 0$ .

?

Problem 1.9.15 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.9 "Functions of Several Variables"

(a)(b)(c)

Define $f(0,0) = 0$ , and put

f(x, y) = x^{2}+y^{2}-2 x^{2} y-\frac{4 x^{6} y^{2}}{\left(x^{4}+y^{2}\right)^{2}}

if $(x, y) \neq (0,0)$ .

?

(a)

Prove, for all $(x, y) \in \mathbb {R}^{2}$ , that

4 x^{4} y^{2} \leq \left(x^{4}+y^{2}\right)^{2}

Conclude that $f$ is continuous.

(b)

For $0 \leq \theta \leq 2 \pi ,-\infty < t < \infty$ , define

g_{\theta }(t) = f(t \cos \theta , t \sin \theta )

Show that $g_{\theta }(0) = 0, g_{\theta }^{\prime }(0) = 0, g_{\theta }^{\prime \prime }(0) = 2$ . Each $g_{\theta }$ has therefore a strict local minimum at $t = 0$ .

In other words, the restriction of $f$ to each line through $(0,0)$ has a strict local minimum at $(0,0)$ .

(c)

Show that $(0,0)$ is nevertheless not a local minimum for $f$ , since $f\left(x, x^{2}\right) = -x^{4}$ .

Problem 1.9.21 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.9 "Functions of Several Variables"

(a)(b)

Define $f$ in $\mathbb {R}^{2}$ by

f(x, y) = 2 x^{3}-3 x^{2}+2 y^{3}+3 y^{2}

?

(a)

Find the four points in $\mathbb {R}^{2}$ at which the gradient of $f$ is zero. Show that $f$ has exactly one local maximum and one local minimum in $\mathbb {R}^{2}$ .

(b)

Let $S$ be the set of all $(x, y) \in \mathbb {R}^{2}$ at which $f(x, y) = 0$ . Find those points of $S$ that have no neighborhoods in which the equation $f(x, y) = 0$ can be solved for $y$ in terms of $x$ (or for $x$ in terms of $y$ ). Describe $S$ as precisely as you can.

Problem 1.9.22 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.9 "Functions of Several Variables"

Give a similar discussion for

f(x, y) = 2 x^{3}+6 x y^{2}-3 x^{2}+3 y^{2}

?

Problem 1.9.31 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.9 "Functions of Several Variables"

Suppose $f \in \mathscr {C}^{(3)}$ in some neighborhood of a point $\mathbf{a} \in \mathbb {R}^{2}$ , the gradient of $f$ is 0 at a, but not all second-order derivatives of $f$ are 0 at a. Show how one can then determine from the Taylor polynomial of $f$ at a (of degree 2) whether $f$ has a local maximum, or a local minimum, or neither, at the point a.

Extend this to $\mathbb {R}^{n}$ in place of $\mathbb {R}^{2}$ .

?

Local extrema of functions of several variables

References