AM GM help (1 Viewer)

TL1998 · Jun 24, 2014

Sketch y = 2^x, then use the fact that the chord AB (where A(a,2^a) and B(b,2^b))lies above the curve to show that the geometric mean of two distinct positive numbers is less than their arithmeitc mean

RealiseNothing · Jun 24, 2014

The chord lies above the curve right.

Take the point $(\frac{a+b}{2}, 2^{\frac{a+b}{2}})$

Now the y co-ordinate on the chord at this point is $\frac{2^a+2^b}{2}$

This is greater than the y co-ordinate on the curve, so:

$\frac{2^a+2^b}{2} \geq 2^{\frac{a+b}{2}}$

But $2^{\frac{a+b}{2}} = \sqrt{2^a2^b}$

So:

$\frac{2^a+2^b}{2} \geq \sqrt{2^a2^b}$

Let $x=2^a$ and $y=2^b$ and we are done.

TL1998 · Jun 25, 2014

Okay good. Had the same solution but wasn't sure whether it was valid.

AM GM help (1 Viewer)

TL1998

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RealiseNothing

what is that?It is Cowpea

TL1998

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