Final answer:
To find the closest points to the origin on the surface y² = 9 + xz, the distance function d² = x² + y² + z² is minimized by substituting y² with the given surface equation and setting the partial derivatives with respect to x and z to zero.
Step-by-step explanation:
To find the points on the surface y² = 9 + xz that are closest to the origin, we need to minimize the distance from a point (x,y,z) on the surface to the origin. The distance squared of any point from the origin in 3-dimensional space is d² = x² + y² + z². However, since y² is given as a function of x and z, we can substitute y² in the expression for d², yielding the new function d² = x² + 9 + xz + z² that we want to minimize.
To find the minimum, we can take the partial derivatives of d² with respect to x and z and set them to zero. Solving the resulting system of equations will give us the coordinates of the closest points to the origin. The partial derivatives are 2x + z (with respect to x) and x + 2z (with respect to z), leading to the simultaneous equations x + 2z = 0 and 2x + z = 0. Solving these will provide the required coordinates.