>>11
All real numbers between 0 and 1 have a decimal expansion, and thus, are decimal numbers. "Feeling" that something is right isn't the same as a rigorous proof.
Furthermore, calculus is a subset of algebra.
>>15
Phail. The minimum is 2 at a = 1.
>>1
The arithmetic mean - root mean square inequality states that for any x,y >0,
[(x
2 + y
2)/2]
1/2 >= [x+y]/2, with equality iff x = y.
Thus,
(a+1/a)
2 + (b+1/b)
2 >= 2[(a+1/a+b+1/b)/2]
2 = (1/2)[1 + 1/a + 1/b]
2 = (1/2)[1 + (b+a)/(ab)]
2 = (1/2)[1 + 1/(ab)]
2
ab = a(1-a) = a - a
2 = 1/4 - (a - 1/2)
2 <= 1/4
(1/2)[1 + 1/(ab)]
2 >= (1/2)[1 + 1/(1/4)]
2 = (1/2)[1 + 4]
2 = (1/2)5
2 = 25/2