For any acute angle , prove that

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- Jun 19th 2009, 04:04 AMgreat_mathInequalities(3):
For any acute angle , prove that

- Jun 19th 2009, 04:56 AMalexmahone
Using Jensen's inequality for the function ,

- Jun 19th 2009, 04:57 AMmalaygoel
Please check the question....I think it is incorrect.

.........my mistake ::::Confused A and a

How to delete the post??? - Jun 19th 2009, 05:02 AMpankaj
Let

,since for

Let

for

For ,we have and thus

Therefore, for all .

Thus the function y=f(x) is concave downwards for all .

- Jun 19th 2009, 05:06 AMpankaj
- Jun 19th 2009, 05:12 AMalexmahone
- Jun 19th 2009, 07:08 PMmalaygoel
- Jun 20th 2009, 09:00 AMpankaj
Consider the three points on the curve and . When joined they form a triangle which obviously lies below the curve .Now centroid of this triangle is inside the triangle and thus centroid is also below the curve.

Co-ordinates of centroid= .

The corresponding point on the curve with the same -coordinate will be

Since this point is above the centroid .Therefore,

Hey,you can try plotting and draw a triangle and check for yourself - Jun 20th 2009, 06:38 PMmalaygoel