Answer
To maximize his satisfaction, Antonio should purchase $\frac{5}{3}$ hamburgers and $\frac{5}{2}$ french fries.
Work Step by Step
We have the function $U\left( {{x_1},{x_2}} \right) = \sqrt {{x_1}{x_2}} $, where ${x_1}$ and ${x_2}$ are the numbers of hamburgers and french fries purchased by Antonio. Our task is to maximize $U$ subject to the constraint $g\left( {{x_1},{x_2}} \right) = \frac{3}{2}{x_1} + {x_2} - 5 = 0$.
Step 1. Write out the Lagrange equations
Using Theorem 1, the Lagrange condition $\nabla U = \lambda \nabla g$ yields
$\left( {\frac{{{x_2}}}{{2\sqrt {{x_1}{x_2}} }},\frac{{{x_1}}}{{2\sqrt {{x_1}{x_2}} }}} \right) = \lambda \left( {\frac{3}{2},1} \right)$
So, the Lagrange equations are
(1) ${\ \ \ \ }$ $\frac{{{x_2}}}{{2\sqrt {{x_1}{x_2}} }} = \frac{3}{2}\lambda $, ${\ \ \ }$ $\frac{{{x_1}}}{{2\sqrt {{x_1}{x_2}} }} = \lambda $
Step 2. Solve for $\lambda$ in terms of $x$ and $y$
Since $\left( {0,0} \right)$ does not satisfy the constraint, we have the following cases:
Case 1. ${x_1} = 0$, ${x_2} \ne 0$
Substituting ${x_1} = 0$ in the constraint gives ${x_2} = 5$. So, the critical point is $\left( {0,5} \right)$.
Case 2. ${x_1} \ne 0$, ${x_2} = 0$
Substituting ${x_2} = 0$ in the constraint gives $\frac{3}{2}{x_1} - 5 = 0$. So, ${x_1} = \frac{{10}}{3}$.
So, the critical point is $\left( {\frac{{10}}{3},0} \right)$.
Case 3. ${x_1} \ne 0$ and ${x_2} \ne 0$
In this case $\lambda \ne 0$. Equation (1) becomes
$\lambda = \frac{{{x_2}}}{{3\sqrt {{x_1}{x_2}} }} = \frac{{{x_1}}}{{2\sqrt {{x_1}{x_2}} }}$
$\frac{{{x_2}}}{3} = \frac{{{x_1}}}{2}$
So, ${x_2} = \frac{3}{2}{x_1}$.
Step 3. Solve for $x$ and $y$ using the constraint
Substituting ${x_2} = \frac{3}{2}{x_1}$ from Step 2 in the constraint gives
$\frac{3}{2}{x_1} + \frac{3}{2}{x_1} - 5 = 0$
$3{x_1} = 5$
${x_1} = \frac{5}{3}$
Using ${x_2} = \frac{3}{2}{x_1}$, we obtain ${x_2} = \frac{5}{2}$.
So, the critical point is $\left( {\frac{5}{3},\frac{5}{2}} \right)$.
Step 4. Calculate the critical values
We evaluate the extreme values at the critical points and list them in the following table:
$\begin{array}{*{20}{c}}
{{\rm{Critical{\ }point}}}&{U\left( {x,y} \right)}\\
{\left( {0,5} \right)}&0\\
{\left( {\frac{{10}}{3},0} \right)}&0\\
{\left( {\frac{5}{3},\frac{5}{2}} \right)}&{\frac{5}{{\sqrt 6 }}}
\end{array}$
From the results in this table we conclude that the maximum value of $U$ subject to the constraint $g\left( {{x_1},{x_2}} \right) = \frac{3}{2}{x_1} + {x_2} - 5 = 0$ is $\frac{5}{{\sqrt 6 }}$.
Thus, to maximize his satisfaction, Antonio should purchase $\frac{5}{3}$ hamburgers and $\frac{5}{2}$ french fries.
