Inequality Proofs, help please.?

Hey guys, I'm having some trouble with a few inequalities I need to prove, any help appreciated. Note if I say >= I mean bigger than or equal to. 1) (a+b)/2 >= sqrt(ab), starting from (sqrt(a) + sqrt(b))^2 >= 0 I can prove this starting from (a-b)^2 >= 0, but when I use the required starting... show more Hey guys, I'm having some trouble with a few inequalities I need to prove, any help appreciated. Note if I say >= I mean bigger than or equal to.

1) (a+b)/2 >= sqrt(ab), starting from (sqrt(a) + sqrt(b))^2 >= 0

I can prove this starting from (a-b)^2 >= 0, but when I use the required starting point my answer is always (a+b)/2 >= -sqrt(ab) i.e. the RHS is negative.

2) (a+b+c+d)/4 >= 4thrt(abcd)

using the arithmetic and geometric mean theorem from the first question. Starting point is given as:

(a+b+c+d)/4 = 1/2[(a+b)/2 + (c+d)/2]


Thanks greatly in advance for help.
Update: Note, thankyou for help but as I said where you have
a + b ≥ -2(√a)(√b)
(a + b)/2 ≥ (√a)(√b)
(a + b)/2 ≥ √(ab)

The minus sign would still remain unless the inequality sign was reversed wouldn't it? Or am I missing something
Update 2: Thank you kindly, haha I worked out part 2 just before I refreshed the page and saw your answer, but thank you very much, great help!
Update 3: Yes I thought that myself but assumed I had missed something. I shall take it up with my teacher.
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