Use lagrange multipliers to find the points on the given surface that are closest to the origin. y2 = 64 + xz

Using Lagrange multipliers, it is found that the points on the surface closest to the origin are (0,-8,0) and (0,8,0).

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The distance of any point (x,y,z) to the origin is:

Thus, the function we want to minimize is:

The constraint is:

Their gradients are:

The minimum distance occurs at points for which:

And

Thus:

, thus:

The system of equations is:

From the second equation, we have ,

Thus, from the first and third equation:

Which is only possible for x = 0 and z = 0.

In the equation for g:

Then, the points are: (0,-8,0) and (0,8,0).

A similar problem is given at link

hiking down a winding trail.

step-by-step explanation:

chiken

step-by-step explanation: