To find the maximum and minimum value of the function f(x, y, z) = xy + z² on the surface x² + y² + z² = 1, we can use the method of Lagrange multipliers.
First, we need to define the Lagrangian function L(x, y, z, λ) = xy + z² + λ(1 - x² - y² - z²).
Next, we need to find the partial derivatives of L with respect to x, y, z, and λ, and set them equal to zero to find the critical points.
∂L/∂x = y - 2λx = 0
∂L/∂y = x - 2λy = 0
∂L/∂z = 2z - 2λz = 0
∂L/∂λ = 1 - x² - y² - z² = 0
Solving these equations simultaneously, we get x = y = ±1/√2, z = 0, and λ = ±1/2√2.
Next, we need to evaluate the function f(x, y, z) = xy + z² at these critical points.
f(1/√2, 1/√2, 0) = 1/2
f(1/√2, -1/√2, 0) = -1/2
Therefore, the maximum value of f(x, y, z) = xy + z² on the surface x² + y² + z² = 1 is 1/2, and the minimum value is -1/2.