Here is the problem you need to solve and then discuss with your classmates!

Question $1$:

A manufacturer's production is modeled by the Cobb$-$Douglas function: $f(x,y)=100x^{\frac{1}{2}}{y}^{\frac{1}{2}}$ where $x$ represents the units of labor $($at $$150$ per unit$)$ and $y$ represents the units of copital $($at $$250$ per unit$).$ The total costs for labor and capital cannot exceed $$50,000.$ Will the manufacturer's maximum production level exceed $16,000$ units?

$($Hint: It's not a regular max$/$min problem $-$ this one has constraints! Hence? Lagrange Multiplier time?$)$