The Ace Novelty company produces two souvenirs: Type A and Type B. The number of Type A souvenirs, x, and the number of Type B souvenirs, y, that the company can produce weekly are related by the equation 2x2 + y − 4 = 0, where x and y are measured in units of a thousand. The profits for a Type A souvenir and a Type B souvenir are $4 and $2, respectively. How many of each type of souvenirs should the company produce to maximize its profit?

The method of Lagrange multipliers can solve this quickly. For objective function f(x, y) and constraint function g(x, y)=0 we can set the partial derivatives of the Lagrangian to zero to find the values of the variables at the extreme of interest.

These functions are ...

[tex]f(x,y)=4x+2y\\g(x,y)=2x^2+y-4[/tex]

The Lagrangian is ...

[tex]\mathcal{L}(x,y,\lambda)=f(x,y)+\lambda g(x,y)\\\\\text{and the partial derivatives are ...}\\\\\dfrac{\partial \mathcal{L}}{\partial x}=\dfrac{\partial f}{\partial x}+\lambda\dfrac{\partial g}{\partial x}=4+\lambda (4x)=0\ \implies\ x=\dfrac{-1}{\lambda}\\\\\dfrac{\partial \mathcal{L}}{\partial y}=\dfrac{\partial f}{\partial y}+\lambda\dfrac{\partial g}{\partial y}=2+\lambda (1)=0\ \implies\ \lambda=-2[/tex]

[tex]\dfrac{\partial\mathcal{L}}{\partial\lambda}=\dfrac{\partial f}{\partial\lambda}+\lambda\dfrac{\partial g}{\partial\lambda}=0+2x^2+y-4=0\ \implies\ y=4-2x^2\\\\\text{We know $\lambda$, so we can find x and y:}\\\\x=\dfrac{-1}{-2}=0.5\\\\y=4-2\cdot 0.5^2=3.5[/tex]

Since x and y are in thousands, maximum profit is to be had when the company produces ...

500 Type A souvenirs, and 3500 Type B souvenirs