
 
\subsubsection {}
Vy�et�ujte maxima a minima funkce

$x^2+y$ na $M = \{(x,y); 2x^2+y^2 \le 1\}$;

$x+2y$ pro $x^2+y^2 \le 5$;

$xy$ pro $|x|+|y|\le 1$ nebo $x+y=1$;

$(x^2+y)\exp(-(x^4+y^4))$ a $\frac{xy^2}{1+(|x|+|y|)^4}$ a 
$\frac{1+x-y}{\sqrt{1+x^2+y^2}}$ na ${\Bbb R }^2$.


\subsubsection {}
 Co znamen� $f:(M,\rho) \to {\Bbb R }$ m� v $a \in M$ lok�ln� 
extr�m?

Doka�te, �e m�-li $f:(M,\rho) \to {\Bbb R }$ v $a$ lok�ln� maximum, pak pro 
$a \in N \subset M$ m� $g = f\restriction N:N \to {\Bbb R }$ v $a$ lok�ln� 

\subsubsection {}
Zopakujte si pojmy omezen� mno�ina, uzav�en� mno�ina, kompaktn� 
mno�ina a Weierstrassovy v�ty o nab�v�n� extr�m�.

\subsubsection {}
 Najd�te maxima a minima funkce

(a) $f(x,y) = \sin x + \sin x - \sin (x+y)$ na $\{(x,y) \mid x \ge 0, y 
\ge 0, x+y \le 2\pi\}.$

(b) $f(x,y,z) = a^2 x^2 + b^2 x^2 + c^2 x^2 - (a x^2 + b y^2 + c z^2)^2$ 
na $\{(x,y,z) \mid x^2 + y^2 + z^2 = 1\},$ kde $a > b > c > 0.$

\subsubsection {}
Najd�te maxima a minima funkce

(a) $x - 2 y + 2 z$ za podm�nky (s vazbou) $x^2 + y^2 + z^2 = 9.$

(b) $xyz$ za podm�nek $x+y+z=5$ a $xy+yz+zx=8.$

(c) $\cos ^2 x +\cos ^2 y$ za podm�nky $y - x = \frac{\pi}{4}.$

\subsubsection {}
Definujeme $f(x,y) = x^2 + y^2 + x + y^3$ a $K = \{(x,y)\mid 
g(x,y) = x + y^2 = 0\}$. Doka�te:

(a) $(0,0)$ je stacion�rn� bod Lagrangeovy funkce $L = f + \lambda g$, 
kde $\lambda = -1$. 

(b) $f''(0,0)$ je pozitivn� definitn�.

(c) $f\restriction K$ nem� v bod� $(0,0)$ lok�ln� extr�m.

V�imn�te si, �e $L\restriction K = f\restriction K$, tedy m�-li $L$ 
lok�ln� extr�m v $(0,0)$, m� ho i $L\restriction K = f\restriction K$. K 
tomu v�ak sta�� podstatn� m�n� (viz nap�. Jarn�k: Dif.po�et II).

Najd�te v�echny lok�ln� extr�my $f$ na $K$.


\subsubsection {}
Nalezn�te absolutn� extr�my funkce na mno�in�
$M$

$f\left(x,y,z\right) = \left(x + y\right)^2 + \left(x -
y\right)^2 +
z; M = [-1;1] \times [-1;1] \times
   [-1;1];$

$f\left(x,y\right) = {x \over a} + {y \over b}, a > 0, b
> 0;
   M = \{\left(x,y\right); x^2 + y^2 \leq 1 \};$

$f\left(x,y,z\right) = x^2 + y^2 + z^2 + 2x + 4y - 6z; M
= {\Bbb
R}^3;$

$f\left(x,y\right) = \left(x^2 + y^2\right)e^{-\left(x^2
+
y^2\right)}; M = {\Bbb R}^2;$

$f\left(x,y,z\right) = x^2 + y^2 + z^2; M=
\{\left(x,y,z\right); {x^2
\over a^2} + {y^2 \over
b^2} + {z^2 \over c^2} \leq 1\}; a > b > c > 0.$

$z\left(x,y\right) = x^2 + y^2,$ $M =
\{\left(x,y\right), x^2 +
4y^2 = 1 \}$

$z\left(x,y\right) = x^2 + 12xy + 2y^2;$ $M = \{
\left(x,y\right),
4x^2 + y^2 = 1 \}$

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

$f\left(x,y,z\right) = xyz;$ $M = \{ \left(x,y,z\right),
x^2 + y^2 +
z^2 = 1 \}$

$u\left(x,y,z\right) = \sin {x} \sin {y} \sin {z};$ $ M
=
\{\left(x,y,z\right); x + y +z = {\pi
   \over 2}, x > 0, y > 0, z >0 \}$

$z\left(x,y\right) = x^2 + 12xy + 2y^2; \, M = \{
\left(x,y\right), 4x^2 + y^2
= 1 \}$

$u\left(x,y,z\right) = x - 2y + 2z; \, M = \{ \left(x,y,z\right),
x^2 + y^2 +
z^2 = 1,
x+y+z = 0 \}$


$f\left(x,y\right) = x^2 - xy + y^2, \qquad
M = \{\left(x,y\right) \in \Bbb {\Bbb R}^2; \, |x| + |y| \leq
1\}$


$u\left(x,y,z\right) = xy^2z^3;$ $M =
\{\left(x,y,z\right); x + 2y +
3z = a, x >0, y >0 \}
   \mbox{; kde } a > 0.$

 $u\left(x_1, x_2, \dots, x_n\right) = x_1^p + x_2^p +
\dots +
x_n^p;\,
   M = \{ \left(x_1, x_2, \dots, x_n\right); x_1 + x_2 + \dots +
x_n = a; x_1 >
0,
   \dots, x_n > 0 \};$
   kde $a>0, p > 1.$















\subsubsection {}
Ur�ete rozm�ry vodn� n�dr�e ve tvaru kv�dru o objemu
$32 m^3$ tak,
aby
dno a st�ny m�ly dohromady nejmen�� povrch.




\subsubsection {}
Pomoc� v�ty o multiplik�torech doka�te n�sleduj�c�
nerovnost:

   $$ {x^n + y^n \over 2} \ge \left({x+y \over 2}\right)^n;
\mbox{ kde } n \ge
1,
      x \ge 0, y \ge 0.$$




\subsubsection {}
V n�sleduj�c�ch �loh�ch zjist�te $\sup\limits_{z
\in
M}{f\left(z\right)}$ a
$\inf\limits_{z \in M}{f\left(z\right)}$ a vy�et�ete,
zda t�chto hodnot $f$ na $M$ nab�v�.

$f\left(x,y,z\right) = xyz; M = \{\left(x,y,z\right)
\in \Bbb {\Bbb
R}^3; \, x^2 + y^2 + z^2 =
1, \, x + y = z = 0\}$

$f\left(x,y\right) = \left(x+y\right)e^{-2x -3y}; M =
\{\left(x,y\right); x > 0, y > 0 \}$
\pr3\right) $f\left(x,y\right) = 2x^2 - xy + y^2; M =
\{\left(x,y\right), |x| +
|y| \le 1\}$

$f\left(x,y\right) = x + y; M = \{\left(x,y\right);
x^3 + y^3 - 2xy
= 0, x \ge 0, y \ge
0 \}$






\subsubsection {}
Nalezn�te lok�ln� extr�my n�sleduj�c�ch funkc� v
uveden�ch oblastech:  $f\left(x,y\right) = x^4 + y^4 - x^2 -2xy - y^2; M =
{\Bbb R}^2$


$f\left(x,y\right) = xy + \frac{50}{x} +
\frac{20}{y}; M =
\{\left(x,y\right); x > 0; y
> 0 \}$

$z\left(x,y\right) = x^2y^3\left(6 - x - y\right); M
=
\{\left(x,y\right); x > 0; y > 0\}$


$f\left(x,y,z\right) = x + {y^2 \over 4x} + {z^2
\over y} + {2
\over z}$,
 $M = \{\left(x,y,z\right); x > 0, y > 0, z > 0 \}.$

$f\left(x,y,z\right)  = x^4 - y^4 - x^2 - 2xy - y^2$
\pr7\right) $h\left(x,y\right) = xy + \frac {50}{x} + \frac
{20}{y} \quad
\left(x > 0, \, y > 0\right).$




\subsubsection {}
Doka�te, �e funkce
$
f\left(x,y\right) = \left(1 + e^y\right)\cos x - y e^y
$
nab�v� lok�ln�ho maxima v nekone�n� mnoha bodech, av�ak
lok�ln�ho minima ani v~jednom bod�.

