Minimize the function s=y given the constraint x^2+y^2+z^2=1. Find the values of x, y and z so that s is minimal.
To minimize the function s=y, given the constraint x^2+y^2+z^2=1, we can use the method of Lagrange multipliers.
First, we define the Lagrangian function L, which is the function we want to minimize together with the constraint multiplied by a Lagrange multiplier λ:
L = y - λ(x^2 + y^2 + z^2 - 1)
To find the values of x, y, and z that minimize s, we need to find the points where the gradient of L is equal to zero, which gives us the critical points.
∇L = (∂L/∂x, ∂L/∂y, ∂L/∂z, ∂L/∂λ) = (0, 0, 0, 0)
Taking partial derivatives with respect to each variable, we get:
∂L/∂x = 0 - 2λx = -2λx = 0
∂L/∂y = 1 - 2λy = 0
∂L/∂z = 0 - 2λz = -2λz = 0
∂L/∂λ = x^2 + y^2 + z^2 - 1 = 0
From the first equation, we can see that either λ = 0 or x = 0. If λ = 0, then from the second equation, y = 1/2. If x = 0, then from the third equation, z = 0.
If λ = 0 and x = 0, the fourth equation gives us y^2 + z^2 = 1. This represents a circle in the yz-plane.
If λ = 0 and y = 1/2, the fourth equation gives us 1/4 + z^2 = 1, which simplifies to z^2 = 3/4. This represents two points on the z-axis.
Therefore, the critical points are:
1) (0, 1/2, √3/2)
2) (0, 1/2, -√3/2)
To find the minimum value of s, we substitute these points into the function s=y:
s1 = 1/2
s2 = 1/2
Therefore, the minimal value of s is 1/2, and it occurs at the points (0, 1/2, √3/2) and (0, 1/2, -√3/2).