Экстремумы функций

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

вдоль каждой прямой, проходящей через точку $(0 ; 0)$, имеет в этой точке минимум;

не имеет минимума в точке $(0 ; 0)$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$u = x y, x+y-1 = 0$;

$u = x^{2}+y^{2}, 3 x+2 y-6 = 0$;

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

$u = x y^{2}, x+2 y-1 = 0$;

$u = \cos^{2} x+\cos^{2} y, x-y-\pi / 4 = 0$.

$u = x y$;

$u = x^{2}+y^{2}$

$u = x^{2}-y^{2}$

$u = x y^{2}$.

$u = 5-3 x-4 y, x^{2}+y^{2} = 25$;

$u = 1-4 x-8 y, x^{2}-8 y^{2} = 8$;

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

$u = 2 x^{2}+12 x y+y^{2}, x^{2}+4 y^{2} = 25$;

$u = x / a+y / b, x^{2}+y^{2} = r^{2}, r > 0$.

$u = (x-1)^{2}+(y+1)^{2}$, а) $x^{2}+y^{2}-2 x y = 0$, б) $x-y = 0$;

$u = x^{4}+y^{4},(x-1)^{3}-y^{2} = 0$.

$u = 1+\frac{1}{x}+\frac{1}{y}, \frac{1}{x^{2}}+\frac{1}{y^{2}} = \frac{1}{8}$;

$u = \ln x y, x^{3}+x y+y^{3} = 0$.

$u = 2 x^{2}+3 y^{2}+4 z^{2}, x+y+z = 13$

$u = x y^{2} z^{3}, x+y+z = 12, x > 0, y > 0, z > 0$;

$u = x^{2} y^{3} z^{4}, 2 x+3 y+4 z = 18, x > 0, y > 0, z > 0$

$u = \sin x \sin y \sin z, x+y+z = \pi / 2, x > 0, y > 0, z > 0$;

$u = x-2 y+2 z, x^{2}+y^{2}+z^{2} = 9$

$u = x-y+2 z, x^{2}+y^{2}+2 z^{2} = 16$;

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

$u = x y+2 x z+2 y z, x y z = 108$;

$u = x^{2}+y^{2}+z^{2}, x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1, a > b > c > 0$;

$u = x+y+z, a / x+b / y+c / z = 1, a > 0, b > 0, c > 0$.

$u = x y z, x+y-z = 3, x-y-z = 8$;

$u = x y z, x y+y z+z x = 8, x+y+z = 5$

$u = x y+y z, x^{2}+y^{2} = 2, y+z = 2, y > 0$;

$u = x^{2}+y^{2}+z^{2}, x^{2} / 4+y^{2}+z^{2} = 1, x+y+z = 0$;

$u = (x-1)^{2}+(y-2)^{2}+(z-3)^{2}, x^{2}+y^{2}+z^{2} = 21,3 x+2 y + z = 0$;

$u = x^{2} / 4+y^{2}+z^{2}, x^{2}+y^{2}+z^{2} = 1, x+2 y+3 z = 0$.

$u = \sum_{i = 1}^{n} a_{i} x_{i}^{2}, \sum_{i = 1}^{n} x_{i} = 1, a_{i} > 0$

$u = \sum_{i = 1}^{n} x_{i}^{2}, \sum_{i = 1}^{n} \frac{x_{i}}{a_{i}} = 1, a_{i} > 0$;

$u = \sum_{i = 1}^{n} x_{i}^{\alpha }, \sum_{i = 1}^{n} x_{i} = a, \alpha > 0, a > 0$;

$u = \sum_{i = 1}^{n} \frac{a_{i}}{x_{i}}, \sum_{i = 1}^{n} b_{i} x_{i} = 1, a_{i} > 0, b_{i} > 0, x_{i} > 0$

$u = \prod_{i = 1}^{n} x_{i}^{\alpha_{i}}, \sum_{i = 1}^{n} x_{i} = a, \alpha_{i} > 0, a > 0$;

$u = \sum_{i = 1}^{n} a_{i} x_{i}, \sum_{i = 1}^{n} x_{i}^{2} = 1$.

$u = x y+x+y,-2 \leqslant x \leqslant 2,-2 \leqslant y \leqslant 4$;

$u = x^{2}-x y+y,\left|x\right| \leqslant 2,\left|y\right| \leqslant 3$;

$u = x^{2}+y^{2}-4 x,-2 \leqslant x \leqslant 1,-1 \leqslant y \leqslant 3$;

$u = x^{3}+y^{3}-3 x y, 0 \leqslant x \leqslant 2,-1 \leqslant y \leqslant 2$;

$u = x^{3}+8 y^{3}-6 x y+1,0 \leqslant x \leqslant 2,\left|y\right| \leqslant 1$

$u = x+\left|x-y\right|,\left|x\right| \leqslant 1,\left|y\right| \leqslant 2$

$u = x^{2}-x y+y^{2},\left|x\right|+\left|y\right| \leqslant 1$

$u = (x+y) e^{x y},-2 \leqslant x+y \leqslant 1$.

$u = 1+x+2 y, x+y \leqslant 1, x \geqslant 0, y \geqslant 0$;

$u = x+3 y, x+y \leqslant 6, x+4 y \geqslant 4, y \leqslant 2$;

$u = x^{2}-2 y+3, y-x \leqslant 1, x \leqslant 0, y \geqslant 0$

$u = x^{2}+y^{2}-x y-x-y, x+y \leqslant 3, x \geqslant 0, y \geqslant 0$;

$u = x y(6-x-y), x+y \leqslant 12, x \geqslant 0, y \geqslant 0$;

$u = \sin x+\sin y-\sin (x+y), x+y \leqslant 2 \pi , x \geqslant 0, y \geqslant 0$.

$u = 3+2 x y$, a) $x^{2}+y^{2} \leqslant 1$, б) $4 \leqslant x^{2}+y^{2} \leqslant 9$;

$u = (x-6)^{2}+(y+8)^{2}, x^{2}+y^{2} \leqslant 25$;

$u = x^{2}-y^{2}, x^{2}+y^{2} \leqslant 2 x$

$u = x^{2} y, x^{2}+y^{2} \leqslant 1$

$u = y^{4}-x^{4}, x^{2}+y^{2} \leqslant 9$;

$u = \left(y^{2}-x^{2}\right) e^{1-x^{2}+y^{2}}, x^{2}+y^{2} \leqslant 4$.

$u = x+2 y+3 z, x+y \leqslant 3, x+y \leqslant z, 3 x+3 y \geqslant z, x \geqslant 0$, $y \geqslant 0$

$u = 3 z-y-2 x, x+y \geqslant 2,3 x+y \leqslant 6,0 \leqslant z \leqslant 3, x \geqslant 0$;

$u = x+y+z, x^{2}+y^{2} \leqslant z \leqslant 1$

$u = x^{2}+2 y^{2}+3 z^{2}, x^{2}+y^{2}+z^{2} \leqslant 100$.

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

$u = x^{2}+2 y^{2}+3 z^{2}, x^{2}+y^{2}+z^{2} = 1, x+2 y+3 z = 0$;

$u = 3 x^{2}+4 y^{2}+5 z^{2}+4 x y-4 y z, x^{2}+y^{2}+z^{2} = 1$;

$u = y^{2}+4 z^{2}-2 x y-2 x z-4 y z, 2 x^{2}+3 y^{2}+6 z^{2} = 1$.

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

$u = \left|x+y\right|-\sqrt{1-x^{2}-y^{2}}$;

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

$u = \left(x^{2}+y^{2}+z^{2}\right) e^{-\left(x^{2}+2 y^{2}+3 z^{2}\right)}$.

$y = x^{2}, x-y-5 = 0$;

$x^{2}-y^{2} = 3, y-2 x = 0$;

$9 x^{2}+4 y^{2} = 36,3 x+y-9 = 0$;

$2 x^{2}-4 x y+2 y^{2}-x-y = 0,9 x-7 y+16 = 0$.

поверхность его равна $S$;

сумма длин ребер равна $a$.

в полусферу радиуса $R$;

в прямой круговой конус, радиус основания которого равен $r$, а высота $H$;

в эллипсоид, полуоси которого равны $a, b, c$;

в сегмент эллиптического параболоида

$z / c = x^{2} / a^{2}+y^{2} / b^{2}, \quad z = h, \quad a > 0, \quad b > 0, \quad c > 0, \quad h > 0$.

$\left\{ \begin{aligned} A x+B y+C z & = 0, \\ x^{2} / a^{2}+y^{2} / b^{2} & = 1 ;\end{aligned}\right\}$

$\begin{cases} A x+B y+C z = 0, \\ x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2} = 1. \end{cases}$

сумма их кубов наименьшая;

сумма их обратных величин наименьшая.

Найти значения суммы.