solve using graph

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If A+B+C=Pi , then prove that cos A+cos B+ cos C<3/2, where A, B, C are distinct.

solve using graph

#Mathematics #Calculus

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5 Answers

62

Lokesh Verma ·2010-01-23 00:58:05

Jensen's inequality will help.. (but there is a small issue there wchih i think can be sorted out.... )

http://en.wikipedia.org/wiki/Jensen%27s_inequality

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11

Tush Watts ·2010-01-23 01:11:35

Using Jenson's Inequality ,
If f(x) is such that f '(x) > 0 and f '' (x) < 0 , then

f (m x₂+ n x₁m+n) â‰¥ n f(x₁) + m f(x₂)m+n

We get,
cos ( A + B +C3 ) > cos A + cos B + cos C3

Therefore, 1/2 > cos A + cos B + cos C3
Thus,
cos A+cos B+ cos C<3/2 [Given A+ B+ C = pie]
Hence Proved [1]

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62

Lokesh Verma ·2010-01-23 02:26:13

Tushar this can be proved only if each of A, B and C are less than pi/2

which is infact not given!

so you will have to prove this as well

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11

Tush Watts ·2010-01-23 11:12:48

But sir the equality holds only when A= B= C = pie/3

cos A = cos B = cos C = pie/3 and A + B+ C =pie
or A = pie/3

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Amritansh Bharech ·2010-01-23 11:30:14

what is thios jenson's inequality
i didnt get it
can sum1 explain it in details

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