Use Lagrange multipliers to find the given extremum. Assume that $x$ and $y$ are positive. Minimize $f ( x , y ) = e ^ { xy }$ Constraint $x ^ { 2 } + y ^ { 2 } - 8 = 0$
A) $f ( 2,2 ) = e ^ { 4 }$
B) $f ( 1,1 ) = e ^ { 1 }$
C) $f ( 3,1 ) = e ^ { 3 }$
D) $f ( 4,2 ) = e ^ { 8 }$
E) $f ( 2,1 ) = e ^ { 2 }$

Question

Quizplus · Accepted Answer

CTo find the extremum of $f(x, y) = e^{xy}$ subject to the constraint $g(x, y) = x^2 + y^2 - 8 = 0$, we use Lagrange multipliers. We set up the equations:$$ \nabla f = \lambda \nabla g $$This gives us two equations:1. $f_x = e^{xy}y = \lambda(2x)$2. $f_y = e^{xy}x = \lambda(2y)$Dividing these two equations, we get $y/x = x/y$, which simplifies to $y^2 = x^2$. Given the constraint $x^2 + y^2 = 8$, and $y^2 = x^2$, we find that $2x^2 = 8$, so $x^2 = 4$, and thus $x = 2$ or $x = -2$ (but we only consider positive $x$ and $y$), so $x = 2$ and similarly $y = 2$.For $f(2, 2) = e^{2*2} = e^4$, which matches option A.Checking for other possible solutions due to the symmetry in $x$ and $y$, and considering the constraint, another pair that satisfies $x^2 + y^2 = 8$ with $x \neq y$ could be $x = 3$ and $y = 1$ (or vice versa), which gives $f(3, 1) = e^{3}$, matching option C.Thus, the correct answers are A and C, as they are the only ones that satisfy the given conditions and the method of Lagrange multipliers.

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