F(x, y) = x2 + y2 + 4x − 4y, x2 + y2 ≤ 81 find the extreme values of f on the region described by the inequality.

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are: -8 and 131.91

The given parameters are:

Find the gradient of F(x,y)

Set to 0, to solve for x and y

So, the critical point is:

Also, we have:

Calculate the gradients

Equate the gradients as follows:

So, we have:

Make the subject, in the above equations

Equate the above equations, so we have:

Cross multiply

Subtract xy from both sides

Divide both sides by 2

Substitute in

Divide both sides by 2

Take square roots

Rationalize

Recall that:

So, we have:

The ordered pairs are:

Hence, the points are:

Substitute these values in

Hence, the extreme values of the function are: -8 and 131.91

Read more about extreme values at:

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(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18