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lovemake1

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## Homework Statement

Given 0 <= a <= b

show that,

a <= sqrt(ab) <= (a+ b / 2) <= b

## Homework Equations

a * b <= a^2 / a*b <= a* a

## The Attempt at a Solution

I think i know where I am going but i wanted to make sure if its correct so far.

So we know that the order of least to greatest, 0 -> a -> b

and the first part of inequality states that a <= (Sqrt)ab

so i take the sqrt and squre the left side so it makes a ^2.

The inequality is now a^2 <= ab

and this is true because a < b. and a^2 = a * a

there fore a* a is smaller than a * b.

The second part is [sqrt(ab) <= a+ b/ 2]

Sqrt(ab) smaller or equal to (a + b) / 2.

Solution: square both sides, ab <= [ (a+b)/ 2 ]^2

and this gives 4ab <= (a+b)^2

4ab <= a^2 + 2ab + b^2

and last part

a + b / 2 <= b

multiply by 2 to both sides.

a+b <= 2b

and sinec a < b and

a + b <= b * b

is this correct? am i allowed to do what i just did ? please help me

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