Find the extremum of $f(x, y)$ subject to the given constraint, and state whether it is a maximum or a minimum.
\[
f(x, y)=x y ; 3 x+y=10
\]
Find the Lagrange function $F(x, y, \lambda)$.
\[
F(x, y, \lambda)=\square-\lambda
\]
Answer
\(\boxed{F(x, y, \lambda)=x y-\lambda(3 x+y-10)}\) is the final answer.
Step 1 ：Form the Lagrange function. The Lagrange function is formed by subtracting the product of the Lagrange multiplier (λ) and the constraint from the original function. In this case, the constraint is \(3x + y - 10 = 0\). So, the Lagrange function will be \(f(x, y) - λ(3x + y - 10)\).
Step 2 ：Define the original function and the constraint: \(f = x*y\) and \(g = 3*x + y - 10\).
Step 3 ：Substitute the original function and the constraint into the Lagrange function: \(F = x*y - λ*(3*x + y - 10)\).
Step 4 ：\(\boxed{F(x, y, \lambda)=x y-\lambda(3 x+y-10)}\) is the final answer.
