floymdiT
2021-10-18
Answered

Find the points on the cone $z}^{2}={x}^{2}+{y}^{2$ that are closest to the point (4,2,0)

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tabuordy

Answered 2021-10-19
Author has **91** answers

Using the distance formula we can say that the distance between a general point (x,y,z) and (4,2,0) is given by

Since the points are on the cone we can replace

Consider the following function

Note that D(x,z) will be minimum when f(x,z) is minimum, so we will work with f(x,z) for conveince

To find the minimum value of f(x,y), we will us the second derivative test.

Second derivatives test:

If the second partial derivatives of f are continuous on a disk containing the critical point (a,b) , and

Then we can conclude that:

1)

2)

3)

We can find the local extrema using the Second Derivatives Tes from the section. Note that

is a polynomial function and hence is continuous everywhere in its domain. Thus the theorem applies.

Firstly, we will need the first and second partial derivatives of f:

Now we find the critical points of f. These are the points where both first partial derivatives are 0 at the same time. So let's solve the system of equations

Solve the first equation

Divide both sides by 2

Solve the first equation

Divide both sides by 2

Therefore, the critical point is (2,1)

Now we just neet the value

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