Final answer:
To find the points on the cone closest to the point (2, 2, 0), we need to minimize the distance between a point on the cone and the given point. By substituting the equation of the cone into the expression for the square of the distance and differentiating it, we can find that the points on the cone closest to the given point are the points where x^2 + y^2 = 4 and z = 2.
Step-by-step explanation:
To find the points on the cone closest to the point (2, 2, 0), we can start by expressing the distance between a point on the cone (x, y, z) and the given point (2, 2, 0) in terms of x, y, and z.
Since we want to minimize this distance, we can minimize the square of the distance, which is given by D^2 = (x - 2)^2 + (y - 2)^2 + z^2. Now we can substitute the equation of the cone into the expression for D^2 to eliminate x and y, resulting in D^2 = 4 + z^2 - 4z + 4 - 4z + z^2 = 2z^2 - 8z + 8.
Next, we can find the minimum value of D^2 by differentiating it with respect to z and setting the derivative equal to zero. Differentiating D^2 with respect to z gives d(D^2)/dz = 4z - 8.
Setting this equal to zero, 4z - 8 = 0, and solving for z gives z = 2. Substituting this back into the equation of the cone, we find x^2 + y^2 = 4. Therefore, the points on the cone closest to the given point are the points (x, y, z) where x^2 + y^2 = 4 and z = 2.