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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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