Question #41668

Let $\alpha$ and $\beta$ be the coefficients of $x^5$ and $x^3$ respectively in the binomial expansion of $(1-x^2)^4 (1-x)^3$.

Consider the determinants: $\Delta_1(x,y) = \begin{vmatrix} 1 & 1 & 1 \\ x & y & \alpha \\ x^3 & y^3 & \alpha^3 \end{vmatrix} \quad \text{and} \quad \Delta_2(x,y) = \begin{vmatrix} 1 & 1 & 1 \\ x & y & \beta \\ x^2 & y^2 & \beta^2 \end{vmatrix}$

Let $L_1(x,y) = 0$ be the line obtained from the linear factor of $\Delta_1(x,y)$ that has a negative slope. Let $L_2(x,y) = 0$ and $L_3(x,y) = 0$ be the lines obtained from the linear factors of $\Delta_2(x,y)$ that have zero and infinite slopes, respectively. Assume the leading coefficients of $x$ and $y$ in the expressions for $L_1, L_2,$ and $L_3$ are taken as positive.

Let $R$ be the feasible region in the first quadrant ($x \ge 0, y \ge 0$) satisfying the constraints $L_1(x,y) \le 0$, $L_2(x,y) \le 0$, and $L_3(x,y) \le 0$. Find the maximum value of the objective function $Z = 5x + 7y$ on the region $R$.