Find the absolute maximum and minimum value of the given function in the given r

Question

Find the absolute maximum and minimum value of the given function in the given region.

f(x,y)= sqrt of (x^2 + y^2 -2x + 10); R={(x,y): x^2 + y^2 less than or equal to 4, & y greater than or equal to 0.

I can find the partial derivatives, but I get confused about how to go about finding the max & mins on the boundary. I've looked at several examples and each has a something that I just don't know where it comes from. I need a very careful step by step, why I need to do each step and where the values are coming from. Thank you!!!!!

Explanation / Answer

solution -:

z = sqrt(x^2 + y^2 - 2x + 10)
dz/dx = 2(x-1)/2z
dz/dy = 2y/2z
Obviously the only place both of these partial derivatives can be zero is at the point (1,0). Since it is obvious that large values of x and y will give large values of z, we do not need to do any "second derivative test" to establish that (1,0) is the site of the minimum rather than the maximum.

Along y=0, we would have z = sqrt(x^2-2x+10),
and since that function has only a minimum (no max), its max would occur at one of the endpoints (-2,0) or (2,0). At x=-2, you have z = sqrt(18) and at x=2 you have z = sqrt(10),

Next, to check whether any other point on x^2+y^2=4 can have an "f" value (a "z" value) greater than sqrt(10). If we write z in the form sqrt(4 - 2x + 2), it is plain that the largest value will occur where "x" is most negative.

Final answer:
Minimum value = f(1,0) = 1
Maximum value = f(-2,0) = sqrt(18)

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