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).

-------------------------------

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

54 kilometers per hour

step-by-step explanation:

according to rules:

1 km = 1000 meter

which means that 1 meter = 1/1000 km

1 hour = 3600 second [ as 1 hour = 60 min = 60 *60 seconds ]

which means that 1 second = 1/3600 hour

therefore, 1 kilometer/hour =

thus to convert 15 meter per second to kilometer per hour, we multiply 15 to 3.6 kilometer/hour.

15 meter/second=   which is the required answer.