LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1Fdb01BUExFfn4mbWludXM7fjMufkNWSSYjMjY4O0VOJiMyMDU7flp+TTJ+VX5QTyYjMjY4OyYjMjA1O1RBJiMyNjg7RUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjQ=

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)));

restart: Ur\304\215eme lok\303\241ln\303\255 extr\303\251my n\303\241sleduj\303\255c\303\255 funkce: f:=(x,y)->x^4-3*x^2*y+3*y-y^3; Vykresleme graf, abychom m\304\233li p\305\231edstavu, jak plocha vypad\303\241: plot3d(f(x,y),x=-2..2,y=-2..2,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=-2..2,y=-2..2,axes=boxed,contours=20); Usuzujeme na jeden lok\303\241ln\303\255 minimum a jeden lok\303\241ln\303\255 maximum. \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})}; Ve v\303\275sledku se objevilo RootOf, co\305\276 nazna\304\215uje p\305\231\303\255tomnost v\303\255ce ne\305\276 jedn\303\251 hodnoty (obvykle komplexn\303\255ch \304\215\303\255sel). Na \303\272pravu pou\305\276ijme p\305\231\303\255kaz "allvalues" - pod\303\255vejte se do n\303\241pov\304\233dy a p\305\231\303\255kaz "map", kter\303\275 v na\305\241em p\305\231\303\255pad\304\233 p\305\231\303\255kaz "allvalues" aplikuje na v\305\241echny operandy SB. map(allvalues,SB); P\305\231ekvapiv\304\233 v\303\275sledkem RootOf nen\303\255 nic nov\303\251ho, tud\303\255\305\276 se mno\305\276ina stacion\303\241rn\303\255ch bod\305\257 omezila jen na re\303\241ln\303\251 stacion\303\241rn\303\255 body. Tu\305\241\303\255me, ve kter\303\275ch bodech jsou extr\303\251my a zd\303\241 se, \305\276e body [(-1/2)*sqrt(3),1/2] a [(1/2)*sqrt(3),1/2] budou sedlov\303\251. \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); 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Zjist\303\255me hodnoty fxx a D2 ve stacion\303\241rn\303\255ch bodech: evalf(subs(x=0,y=-1,[fxx,D2])); 1. v\303\275sledek: Tedy v bod\304\233 [0,-1] je ostr\303\251 lok. minimum. evalf(subs(x=0,y=1,[fxx,D2])); 2. v\303\275sledek: v bod\304\233 [0,1] je ostr\303\251 lok. maximum. evalf(subs(x=(-1/2)*sqrt(3),y=1/2,[fxx,D2])); evalf(subs(x=(1/2)*sqrt(3),y=1/2,[fxx,D2])); 3. a 4. v\303\275sledek: v [(-1/2)*sqrt(3),1/2] a [(1/2)*sqrt(3),1/2] nejsou extr\303\251my. Ur\304\215eme nav\303\255c je\305\241t\304\233 funk\304\215n\303\255 hodnoty v extr\303\251mech: evalf(subs(x=0,y=-1,f(x,y)));evalf(subs(x=0,y=1,f(x,y)));

restart: Ur\304\215eme lok\303\241ln\303\255 extr\303\251my n\303\241sleduj\303\255c\303\255 funkce: f:=(x,y)->(x^2+3*y^2)*exp(1-x^2-y^2); Vykresleme graf, abychom m\304\233li p\305\231edstavu, jak plocha vypad\303\241: plot3d(f(x,y),x=-2..2,y=-2..2,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=-2..2,y=-2..2,axes=boxed,contours=15); Usuzujeme na jeden lok\303\241ln\303\255 minimum a dva lok\303\241ln\303\255 maxima. \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 body [1,0] a [-1,0] budou sedlov\303\251. \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=0,[fxx,D2])); 1. v\303\275sledek: Tedy v bod\304\233 [-1,0] nen\303\255 extr\303\251m. evalf(subs(x=0,y=-1,[fxx,D2])); 2. v\303\275sledek: v bod\304\233 [0,-1] je ostr\303\251 lok. maximum. evalf(subs(x=0,y=0,[fxx,D2])); 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 evalf(subs(x=0,y=1,[fxx,D2])); 4. v\303\275sledek: v LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEqYm9kJiMyODM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= [0,1] je ostr\303\251 lok. maximum. 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlFlbjUufnYmIzI1MztzbGVkZWs6fnZ+Ym9kJiMyODM7flsxLDBdfm5lbiYjMjM3O35leHRyJiMyMzM7bS5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn Ur\304\215eme nav\303\255c je\305\241t\304\233 funk\304\215n\303\255 hodnoty v extr\303\251mech: evalf(subs(x=0,y=-1,f(x,y)));evalf(subs(x=0,y=0,f(x,y)));evalf(subs(x=0,y=1,f(x,y)));

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEjZjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ5RidGL0YyRjktRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZlbi9GTkZmbi1JJW1zdXBHRiQ2JUZULUYjNiMtSSNtbkdGJDYkUSIyRidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZgcC1GYW82JUZpbkZjb0Zpby1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GZm82JFEiNEYnRjktRjY2LVEiO0YnRjlGO0ZaRkBGQkZERkZGSEZlbkZN 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