Find the maximum and minimum values of the function f(x,y,z)=x+2y subject to the constraints y^2+z^2=225 and x+y+z=1. Maximum and minimum value is?
To find the maximum and minimum values of the function f(x, y, z) = x + 2y subject to the constraints y^2 + z^2 = 225 and x + y + z = 1, we can use the method of Lagrange multipliers. This method involves finding the critical points of the function by solving the system of equations formed by considering the gradients of both the function and the constraints.
First, let's define the Lagrangian function L(x, y, z, λ, μ) as:
L(x, y, z, λ, μ) = f(x, y, z) + λ(y^2 + z^2 - 225) + μ(x + y + z - 1)
where λ and μ are the Lagrange multipliers. Our goal is to find the values of x, y, z, λ, and μ that satisfy the critical point conditions.
Step 1: Calculating the gradients
∇f = (1, 2, 0)
∇g1 = (0, 2y, 2z)
∇g2 = (1, 1, 1)
Step 2: Setting up the system of equations
∇f = λ∇g1 + μ∇g2
y^2 + z^2 = 225
x + y + z = 1
Putting the gradients in the equation, we have:
(1, 2, 0) = λ(0, 2y, 2z) + μ(1, 1, 1)
This gives us the following three equations:
1 = μ
2 = 2λy + μ
0 = 2λz + μ
Step 3: Solving the system of equations
From the first equation (1 = μ), we can substitute μ in the second and third equations:
2 = 2λy + 1
0 = 2λz + 1
Solving these two equations, we get:
λ = -1/2
y = 1
z = -1/2
Step 4: Substituting the values to find x
Using the equation x + y + z = 1, we can now find x:
x + 1 - 1/2 = 1
x = 1/2
Therefore, the critical point is (x, y, z) = (1/2, 1, -1/2).
Step 5: Evaluating the function at the critical point
Now, substitute the values of x, y, and z into the function f(x, y, z):
f(1/2, 1, -1/2) = (1/2) + 2(1) = 1/2 + 2 = 5/2
So, the function takes on the value 5/2 at its maximum and minimum.
Therefore, the maximum and minimum values of the function f(x, y, z) = x + 2y, subject to the constraints y^2 + z^2 = 225 and x + y + z = 1, are both 5/2.