Maple7 dasturi yordamida Oliy matematika masalalarini yechish

8.14.1-misol. z = x³ + y³ -3xy funktsiyaning ekstremumini tekshiring.
Yech i sh. Berilgan funktsiya uchun z’_x va z’_y har doim mavjud va bu xususiy hosilalarni topamiz

Endi quyidagi sistemani tuzamiz:

bundan x, = 0, x₂ = 1, y₁ = 0, y₂ = 1. Shunday qilib, M₁(0;0) va M₂(1;1) ikkita
statsionar nuqtaga ega bo‘ldik. Endi quyidagilarni topamiz:

U holda

M₁(0;0) nuqtada bo‘lgani uchun bu nuqtada ekstremum yo‘q.
M₂(1;1) nuqtada va bo‘lgani uchun bu nuqtada berilgan
funktsiya lokal minimumga erishadi: z_min=-1
Xususiy hosilalarni topish:
> restart;
> f5:=(x,y)->x^3+y^3-3*x*y;
> f5x:=diff(f5(x,y),x);
> f5y:=diff(f5(x,y),y);
> f5xx:=diff(f5(x,y),x$2);
> f5yy:=diff(f5(x,y),y$2);
> f5xy:=diff(f5(x,y),x,y);

> solvefor({f5x,f5y});
Warning, solvefor is deprecated. Please use solve command.

Statsionar nuqtalarda ekstremumni tekshirish:
1)
> x:=0;y:=0;A5:=f5xx; B5:=evalf(f5xy);C5:=evalf(f5yy); d5:=A5*C5-B5^2;

Ekstremim yo‘q, chunki A5<0, d5<0;
> x:=0; y:=0; f5M1:=x^3+y^3-3*x*y;

1)
> x:=1;y:=1;A5:=f5xx; B5:=evalf(f5xy);C5:=evalf(f5yy); d5:=A5*C5-B5^2;

Funktsiya minimumga ega, chunki A5>0, d5>0;