IESSIAN IN OPTIMIZATION PROBLEMS Optimize the following function, using (a) Cramer's rule for the first-order condition and (b) the Hessian for the second-order condition: \[ y=3 x_{1}^{2}-5 x_{1}-x_{1} x_{2}+6 x_{2}^{2}-4 x_{2}+2 x_{2} x_{3}+4 x_{3}^{2}+2 x_{3}-3 x_{1} x_{3} \]

To optimize the given function, we start with the first-order conditions derived from partial derivatives, then apply Cramer's rule. First, take the gradient of \( y \) with respect to \( x_1, x_2,\) and \( x_3 \). Set each partial derivative to zero to find critical points. Using Cramer's rule helps solve this system of linear equations efficiently. Next, we calculate the Hessian matrix, which consists of the second-order partial derivatives of \( y \). Evaluating the Hessian at the critical points allows us to determine the nature of each point (minima, maxima, or saddle points) based on its definiteness. A positive definite Hessian indicates a local minimum, while a negative definite Hessian suggests a local maximum. Exploring historical optimization techniques reveals that mathematicians like Lagrange and Newton contributed significantly to this field. Lagrange’s method allows for constraints in optimization, while Newton’s method focuses on iteration for finding roots. Understanding their contributions highlights how mathematical optimization combines algebraic and geometric concepts. In real-world applications, optimization techniques are paramount in various fields—be it economics for maximizing profits, engineering for minimizing costs, or machine learning for tuning model parameters. Using an optimization approach leads to better resource allocation and decision-making, proving that these mathematical principles extend beyond textbooks into everyday decision processes.