Question

Minimize f (x, y, z) = xy+xz+yz subject to x² + y² + z² = 2 and z = 1 a) 0

b) 1
c) 2
d) 3

Answer 1

To minimize the function f(x, y, z) given the constraints, convert it into trigonometric form using x = cos(θ) and y = sin(θ), then find where the derivative of the resulting function is zero. The minimum value of the function is 1.

Step-by-step explanation:

The problem asks to minimize f (x, y, z) = xy + xz + yz subject to the constraint x² + y² + z² = 2 and z being fixed at 1. Since z is 1, the expression for f simplifies to f(x, y, 1) = xy + x + y. Now, we are given the constraint of x² + y² + 1² = 2, which simplifies to x² + y² = 1 since z = 1. Recognizing that this is the equation of a circle with radius 1 in the xy-plane, we can use trigonometric substitution to find the minimum of f. Let x = cos(θ) and y = sin(θ), satisfying the circle equation. Substituting these into f gives f(θ) = cos(θ)sin(θ) + cos(θ) + sin(θ). Through differentiation and finding where the derivative equals 0, we can determine the minimum value of f.

Completing this process, the minimum value we find is indeed 1, corresponding to answer option (b).