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Question

# Consider the function $f(x, y)=x y+x+y+100$ subject to the constraint $x y=4$. Solve the system in part (a) to verify that $(x, y)=(-2,-2)$ and $(x, y)=(2,2)$ are solutions.

Solution

Verified

To solve the system, first express $x$ and $y$ from the first two equations:

\begin{align*} & x = y = \dfrac{1}{\lambda-1} \end{align*}

Since $x= y$, the third equation implies:

$x^2 = 4 \implies x = \pm 2$

Hence, the two solutions to the system are $(2,2)$ and $(-2,-2)$.

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