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Tuesday, September 1, 2015

Develep the power of observation

If you're given the two graphs that are plotted in the same Cartesian plane below, what do you notice?



What kind of inequality can you generate from it?

It's obvious that for x>1, the graph of y=x1 is always greater than y=lnx.

Therefore, we can say that lnx<x1 for x>1.

This is a very important discovery and this discovery can help us to prove many HARD and DIFFICULT IMO inequality problem.

But, do you also aware that we could replace x by any suitable replacement we would like and use it to our advantage?

Take for example, we could replace x by x2 such that we have the inequality now defined:

ln(x2)<x21 for x2>1

That is,

ln(x2)<x21 for x>2



Or we could also replace x by x+1 and get:

ln(x+1)<(x+1)1 for x+1>1

which, after cleaning up a bit we get:

ln(x+1)<x for x>0





We can exploit this to prove the inequality below:

(1+1219)(1+1319)<e1218

From the above inequality formula where ln(x+1)<x for x>0, we have:

ln(1+1219)<1219 and

ln(1+1319)<1319

Adding both we obtain:

ln(1+1219)+ln(1+1319)<1219+1319

ln(1+1219)(1+1319)<1219+1319<1219+1219=2219=1218

Exponentiate both sides we get:

(1+1219)(1+1319)<e1218

and we're hence done! :D

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