1. **Problem Statement:** Solve the unconstrained nonlinear multivariable optimization problem: $$f(x_1,x_2) = 2x_1^2 + 4x_2^2 + 3x_1x_2 - 5x_1 - 6x_2 + 7$$ 2. **Formula and Approach:** To find the minimum or maximum of a function of two variables without constraints, we find critical points by setting the partial derivatives to zero: $$\frac{\partial f}{\partial x_1} = 0, \quad \frac{\partial f}{\partial x_2} = 0$$ Then, use the second derivative test (Hessian matrix) to classify the critical points. 3. **Calculate partial derivatives:** $$\frac{\partial f}{\partial x_1} = 4x_1 + 3x_2 - 5$$ $$\frac{\partial f}{\partial x_2} = 8x_2 + 3x_1 - 6$$ 4. **Set derivatives to zero to find critical points:** $$4x_1 + 3x_2 - 5 = 0$$ $$3x_1 + 8x_2 - 6 = 0$$ 5. **Solve the system of linear equations:** Multiply the first equation by 3: $$12x_1 + 9x_2 - 15 = 0$$ Multiply the second equation by 4: $$12x_1 + 32x_2 - 24 = 0$$ Subtract the first from the second: $$\cancel{12x_1} + 32x_2 - 24 - (\cancel{12x_1} + 9x_2 - 15) = 0$$ $$32x_2 - 24 - 9x_2 + 15 = 0$$ $$23x_2 - 9 = 0$$ $$23x_2 = 9$$ $$x_2 = \frac{9}{23}$$ 6. **Substitute $x_2$ back into first equation:** $$4x_1 + 3\left(\frac{9}{23}\right) - 5 = 0$$ $$4x_1 + \frac{27}{23} - 5 = 0$$ $$4x_1 = 5 - \frac{27}{23}$$ $$4x_1 = \frac{115}{23} - \frac{27}{23} = \frac{88}{23}$$ $$x_1 = \frac{88}{23 \times 4} = \frac{88}{92} = \frac{22}{23}$$ 7. **Second derivative test:** Hessian matrix: $$H = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ 3 & 8 \end{bmatrix}$$ Calculate determinant: $$\det(H) = 4 \times 8 - 3 \times 3 = 32 - 9 = 23 > 0$$ Since $\frac{\partial^2 f}{\partial x_1^2} = 4 > 0$ and determinant $> 0$, the critical point is a local minimum. 8. **Final answer:** The function has a local minimum at $$\boxed{\left(x_1, x_2\right) = \left(\frac{22}{23}, \frac{9}{23}\right)}$$