Find the two numbers whose differences is 8 and the sum of their squares is a minimum

Question

Is 11^(1/2) a rational number???
Pls help

Answer 1

1st number --- x
2nd number -- 8-x

sum of squares = x^2 + (8-x)^2
= x^2 + 64 - 16x + x^2
= 2x^2 - 16x + 64
d(sum of squares)/dx = 4x - 16
= 0 for max/min

4x=16
x = 4

the two numbers are both 4

check:
for the answer, sum of squares = 16+16 = 32
let 1st be 3.99, the other 4.01
sum of squares = 3.99^2 + 4.01^2 = 32.002 which is > 32
true for any other combination of x and 8-x

Answer 2

11^(1/2)
= √11

since 11 is not a perfect square,
11^(1/2) is irrational