Final answer:
To find the points on the cone z² = x² + y² that are closest to the point (4,2,0), we need to minimize the distance between these points.
Step-by-step explanation:
To find the points on the cone z² = x² + y² that are closest to the point (4,2,0), we need to minimize the distance between these points. This can be done by finding the gradient vector of the cone and setting it equal to the gradient vector of the distance function between the points:
The gradient vector of the cone is given by ∇(z² - x² - y²) = (2x, -2y, 2z).
The gradient vector of the distance function is given by ∇((x-4)² + (y-2)² + z²).
By setting these two vectors equal and solving the resulting equations, we can find the points on the cone that are closest to the given point.