Question

Use the method of Lagrange multipliers.

Maximize ​f(x,y)=2xy
subject to x+y=10​
simplify: Maxf(x,y)=

Answer 1

Using Lagrange multipliers to maximize the function f(x,y) = 2xy given the constraint x + y = 10, we find that the maximum value of f(x,y) at x = y = 5 is 50.

Step-by-step explanation:

The method of Lagrange multipliers is a strategy used in mathematics to find the local maxima and minima of a function subject to equality constraints. For the given function f(x,y) = 2xy, constrained by x + y = 10, we introduce a Lagrange multiplier λ and set up the following system of equations to solve for x, y, and λ:

• Grad(f) = λ * Grad(g), where g(x, y) = x + y - 10.
• This yields the equations 2y = λ, 2x = λ, and x + y = 10.

Solving this system, we find that x = y = 5. Substituting back into the function f(x,y), we get Max f(x, y) = 2 * 5 * 5 = 50.