Use Lagrange multipliers to find the given extremum. In each case, assume that $x$ and $y$ are positive. Maximize $f ( x , y ) = x y$ Constraint $x + y = 10$
A) $f ( 7,3 ) = 21$
B) $f ( 5,5 ) = 25$
C) $f ( 6,4 ) = 24$
D) $f ( 2,8 ) = 16$
E) $f ( 1,9 ) = 9$

Question

Quizplus · Accepted Answer

The method of Lagrange multipliers involves setting up the equation $\nabla f = \lambda \nabla g$, where $f(x, y) = xy$ is the function to be maximized and $g(x, y) = x + y - 10 = 0$ is the constraint. Solving this system of equations leads to finding that the maximum occurs when $x = y$, given the constraint $x + y = 10$, which implies $x = y = 5$. Substituting these values into $f(x, y)$ gives $f(5, 5) = 25$, which is the maximum value.

Use Lagrange Multipliers to Find the Given Extremum xxx And yyy Are Positive

Use Lagrange Multipliers to Find the Given Extremum $x$ And $y$ Are Positive