Lagrange Multipliers — Question 9

Question 9 – Solution

Strategy. The multiplier equations balance marginal costs against the constraint normal; then a one-variable check rules out other behavior.

Step 1: Multiplier equations With $g=xy$, $$\left\langle 3,2\right\rangle=\lambda\left\langle y,x\right\rangle, \qquad xy=100.$$ Thus $3=\lambda y$ and $2=\lambda x$. Eliminating $\lambda$ gives $$3x=2y, \qquad y=\frac 32x.$$ The positive constraint then yields $$x^2=\frac{200}{3},$$ so $$\boxed{x=\frac{10\sqrt 6}{3},\qquad y=5\sqrt 6}.$$

Step 2: Cost and interpretation $$\boxed{C_{\min}=3\left(\frac{10\sqrt 6}{3}\right)+2(5\sqrt 6) =20\sqrt 6}.$$ The relation $3/y=2/x=\lambda$ says the cost-gradient components are proportional to the production-constraint gradient components.

Step 3: Global verification Using $y=100/x$ for $x>0$, $$C(x)=3x+\frac{200}{x}\longrightarrow\infty$$ as $x\to 0^+$ or $x\to\infty$. Its only critical point is the one found above, and $C''(x)=400/x^3>0$. Hence it is the unique absolute constrained minimum.