MathProblemsBank

2.11.9 Function extrema

Problem: Function \( f\left(x_{1}, x_{2}, x_{3}\right) \) and its constraints are given. a) Compose the Lagrange function; b) Determine the stationary point of the Lagrange function; c) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{2}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=15 x_{1}^{2}+7 x_{2}^{2}+10 x_{3}^{2}+20 x_{1} x_{2}+ \\ +x_{1} x_{3}+3 x_{2} x_{3}+10 x_{1}-25 x_{2}-20 x_{3}, \\ 11 x_{1}+20 x_{2}+10 x_{3}=300, \\ 21 x_{1}+60 x_{2}+x_{3}=270 . \end{array} \]