USING OPTIMIZATION, two numbers have a sum of 20,

What two numbers squared add up to 20 and get the maximum sum value possible. Please show all of you work. I got this so far... x+y = 20 ...... x^2+y^2 = T y = 20 - x x^2 + (20-x)^2 What do I do after this? Please and Thanks!

We want the minimum sum of x + y with respect to:
x^2 + y^2 = 20.

If we solve x^2 + y^2 = 20 for y, we get:
y = ±√(20 - x^2).

Now, we want to minimize x + y. Intuitively, the negative value for y should do this, so pick the negative value for y:
y = -√(20 - x^2).

Now, we want to minimize:
S = x + y = x - √(20 - x^2) = x - (20 - x^2)^(1/2)

Differentiating S yields:
dS/dx = 1 + (1/2)(-2x)(20 - x^2)^(1/2 - 1)
= 1 - x/√(20 - x^2).

Setting dS/dx = 0:
1 - x/√(20 - x^2) = 0
==> x/√(20 - x^2) = 1
==> x = √(20 - x^2)
==> x^2 = 20 - x^2, by squaring both sides
==> x^2 = 10
==> x = ±√10.

Again, we want to minimize x + y, so pick the negative value for x:
x = -√10.

From y = -√(20 - x^2), y = -√(20 - 10) = -√10. Therefore, the required minimum sum is:
-√10 - √10 = -2√10.

I hope this helps!