Let x and y be real numbers such that x^2 + y^2 = 1. Find the maximum value of 2x - 5y.
Let x and y be real numbers such that x^2 + y^2 = 1. What is the maximum value of 2x - 5y?
To find the maximum value of 2x - 5y, we can use the concept of optimization.
We have the constraints x^2 + y^2 = 1, which means that the point (x, y) lies on the unit circle. Our goal is to maximize the expression 2x - 5y.
One way to solve this problem is by using the method of Lagrange multipliers. We need to find the critical points of the function f(x, y) = 2x - 5y subject to the constraint g(x, y) = x^2 + y^2 - 1 = 0.
The Lagrange function L(x, y, λ) is defined as L(x, y, λ) = f(x, y) - λ * g(x, y), where λ is the Lagrange multiplier.
Taking partial derivatives of L with respect to x, y, and λ, we get:
∂L/∂x = 2 - 2λx
∂L/∂y = -5 - 2λy
∂L/∂λ = x^2 + y^2 - 1
Setting these partial derivatives to zero, we have:
2 - 2λx = 0 ---- (1)
-5 - 2λy = 0 ---- (2)
x^2 + y^2 - 1 = 0 ---- (3)
From equation (2), we can solve for λ:
-5 - 2λy = 0
-2λy = 5
λ = -5/(2y)
Substituting this value of λ into equation (1), we can solve for x:
2 - 2( -5/(2y) )x = 0
2 + 5x/y = 0
x = -2y/5
Now, substituting these values of x and λ into equation (3), we have:
(-2y/5)^2 + y^2 - 1 = 0
4y^2/25 + y^2 - 1 = 0
(29/25)y^2 = 1
y^2 = 25/29
y = ±√(25/29)
If y = √(25/29), then x = -2(√(25/29))/5
If y = -√(25/29), then x = -2(-√(25/29))/5
Now, substitute the values of x and y into the expression 2x - 5y to find the maximum value:
2x - 5y = 2(-2(√(25/29))/5) - 5(√(25/29))
= -4(√(25/29))/5 - 5(√(25/29))
= -29√(25/29)/5
Therefore, the maximum value of 2x - 5y is -29√(25/29)/5.
y = +/- sqrt(1-x^2)
if y is to be real, x must be
x^2 </= 1
-1 < x < +1
similarly y ranges from -1 to + 1
so our function 2 x - 5y
is maximum when x is 1 and y is -1
2 + 5 = 7
but, at (1,-1), x^2+y^2 = 2
The maximum occurs when
x = 2/√29
y = -5/√29