№05_001
Пункт №1
\(u'_{x}=2x+y-12\), \(u'_{y}=x+2y-3\).
\(
\left\{\begin{aligned}
& 2x+y-12=0;\\
& x+2y-3=0.
\end{aligned}\right.
\)
\left\{\begin{aligned}
& 2x+y-12=0;\\
& x+2y-3=0.
\end{aligned}\right.
\)
Стационарная точка: \(M(7;-2)\).
\(u''_{xx}=2\), \(u''_{xy}=1\), \(u''_{yy}=2\).
\(\Delta=u''_{xx}\cdot{u''_{yy}}-\left(u''_{xy}\right)^2=3\)
Так как \(u''_{xx}\gt{0}\) и \(\Delta\gt{0}\), то \(M\) - точка минимума.
\(u_{\min}=u(7;-2)=-39\).