<Text-field style="Heading 1" layout="Heading 1">\303\232kol \304\215\303\255slo 6 - lok\303\241ln\303\255 extr\303\251my funkce v\303\255ce prom\304\233nn\303\275ch</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Grafy funkc\303\255</Text-field>

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

[0,0] nen\303\255 extr\303\251m, [6,18] je lok\303\241ln\303\255 maximum

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEjajJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RidGL0YyLUYjNiQtSSNtbkdGJDYkUSIyRidGOUY5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlxvLUZQNiUtRiw2JVEieUYnRi9GMi1GIzYkLUZYNiRRIjNGJ0Y5RjlGZW4tRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEncGxvdDNkRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNjZGKy1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnRlItRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIRltvRl1vRmVvLUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIRltvL0ZORmhwRmVvRmNwRmBvRltxRl5xRmVvRmFxRmVvRmNwLUYsNiVRJWF4ZXNGJ0YvRjJGW3EtRiw2JUY6Ri9GMkY5RjktRjY2LVEiO0YnRjlGO0ZmcEZARkJGREZGRkhGZ3BGTS8lK2V4ZWN1dGFibGVHRj1GOQ==

[0,0] nen\303\255 extr\303\251m

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

[0,0] je lok\303\241ln\303\255 (glob\303\241ln\303\255) maximum

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

Osa OY je mno\305\276inou lok\303\241ln\303\255ch minim\305\257

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

[0,0] nen\303\255 extr\303\251m, [1,1] je lok\303\241ln\303\255 minimum

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

[0,0] nen\303\255 extr\303\251m, [-4,-2] je lok\303\241ln\303\255 maximum

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

[1,0], [-1,0], [0,1], [0,-1] nejsou extr\303\251my, [1/sqrt(2e),1/sqrt(2e)], [-1/sqrt(2e),-1/sqrt(2e)] jsou lok\303\241ln\303\255 minima, [-1/sqrt(2e),1/sqrt(2e)], [1/sqrt(2e),-1/sqrt(2e)] jsou lok\303\241ln\303\255 maxima

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

[0,0] je lok\303\241ln\303\255 maximum, [1/2,1], [1/2,-1], [-1/2,1], [-1/2,-1] jsou lok\303\241ln\303\255 minima, [0,1], [0,-1], [1/2,0], [-1/2,0] nejsou extr\303\251my

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

[1,-1] je lok\303\241ln\303\255 minimum

<Text-field style="Heading 2" layout="Heading 2">Lok\303\241ln\303\255 extr\303\251my funkce v\303\255ce prom\304\233nn\303\275ch</Text-field>

restart:

Ur\304\215eme lok\303\241ln\303\255 extr\303\251my n\303\241sleduj\303\255c\303\255 funkce:

f:=(x,y)->x*y*exp(-(x^2+y^2)/2);

Vykresleme graf, abychom m\304\233li p\305\231edstavu, jak plocha vypad\303\241:

plot3d(f(x,y),x=-3..3,y=-3..3,style=patch,axes=framed,grid=[40,30], labels=[x,y,z]);

Zn\303\241zorn\303\255me si vrstevnice:

with(plots):contourplot(f(x,y),x=-3..3,y=-3..3,axes=boxed);

Usuzujeme na dva lok\303\241ln\303\255 maxima a dva minima. \303\232vahu ov\303\251\305\231\303\255me v\303\275po\304\215tem.

Nejprve ur\304\215\303\255me stacion\303\241rn\303\255 body (mn. stac. bod\305\257 ozna\304\215\303\255me SB):

SB:={solve({diff(f(x,y),x),diff(f(x,y),y)},{x,y})};

Tu\305\241\303\255me, ve kter\303\275ch bodech jsou extr\303\251my a zd\303\241 se, \305\276e bod [0,0] bude sedlov\303\275. \303\232vahy podpo\305\231ime v\303\275po\304\215tem druh\303\275ch derivac\303\255:

fxx:=factor(diff(f(x,y),x,x));

fxy:=factor(diff(f(x,y),x,y));

fyy:=factor(diff(f(x,y),y,y));

D2:=factor(fxx*fyy-fxy^2);

Zjist\303\255me hodnoty fxx a D2 ve stacion\303\241rn\303\255ch bodech:

evalf(subs(x=1,y=1,[fxx,D2]));

1. v\303\275sledek: Tedy v bod\304\233 [1,1] je ostr\303\251 lok. maximum.

evalf(subs(x=-1,y=-1,[fxx,D2]));

2. v\303\275sledek: V [-1,-1] je tak\303\251 ostr\303\251 lok. maximum.

evalf(subs(x=-1,y=1,[fxx,D2]));

evalf(subs(x=1,y=-1,[fxx,D2]));

3. a 4. v\303\275sledek: V [-1,1] a [1,-1] jsou ostr\303\241 lok. minima.

evalf(subs(x=0,y=0,[fxx,D2]));

5. v\303\275sledek: Tedy v [0,0] extr\303\251m nenast\303\241v\303\241.

Ur\304\215eme nav\303\255c je\305\241t\304\233 funk\304\215n\303\255 hodnoty v extr\303\251mech:

evalf(subs(x=1,y=1,f(x,y)));evalf(subs(x=-1,y=-1,f(x,y)));

evalf(subs(x=-1,y=1,f(x,y)));evalf(subs(x=1,y=-1,f(x,y)));