2.11.8 Function extrema

Problem: A function of three variables is given: \( f\left(x_{1}, x_{2}, x_{3}\right) \). a) Find the stationary point of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), calculate the value of the function and the gradient at it. (Indicate the method for solving the system of equations). b) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=3 x_{1}^{2}+8 x_{2}^{2}+13 x_{3}^{2}-4 x_{1} x_{2}- \\ -8 x_{1} x_{3}-6 x_{2} x_{3}-55 x_{1}-18 x_{2}-21 x_{3} . \end{array} \]