The Lagrangian function corresponding to the following constrained optimization problem: Maximize: $f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160$, subject to $X_{1}+X_{2}=10$ is
A) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}\right)$
B) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda .+\left(X_{1}+X_{2}\right)$
C) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160+\lambda \cdot\left(X_{1}+X_{2}-10\right)$
D) $f\left(X_{1}, X_{2}, \lambda\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160-\lambda \cdot\left(X_{1}+X_{2}+10\right)$

Question

Quizplus · Accepted Answer

The Lagrangian function for a constrained optimization problem includes the objective function and the constraint(s) multiplied by lambda ($\lambda$), the Lagrange multiplier. The correct form for the Lagrangian when the constraint is $X_1 + X_2 = 10$ is to subtract 10 from the sum $X_1 + X_2$ and multiply by $\lambda$, as shown in option C.

The Lagrangian Function Corresponding to the Following Constrained Optimization Problem f(X1,X2)=−2X12+12X1−2X1X2−X22+4X2−160f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160f(X1​,X2​)=−2X12​+12X1​−2X1​X2​−X22​+4X2​−160

The Lagrangian Function Corresponding to the Following Constrained Optimization Problem $f\left(X_{1}, X_{2}\right)=-2 X_{1}^{2}+12 X_{1}-2 X_{1} X_{2}-X_{2}^{2}+4 X_{2}-160$