Papers of Combinatorics

Question:

$i) P r o v e t h a t t h e g r a p h o f y = \log_{e} x i s c o n c a v e d o w n f o r a l l x > 0 . i i) S u p p o s e 0 < a < b a n d c o n s i d e r t h e p o i n t s, A (a_{1} \log_{e} a) a n d B (b, \log_{e} b), o n t h e g r a p h o f y = \log_{e} x . F i n d t h e c o o r d i n a t e s o f t h e p o i n t P t h a t d i v i d e s t h e l i n e s e g m e n t A B i n t e r n a l l y i n t h e r a t i o 2 : 1 . i i i) H e n c e p r o v e : \frac{1}{3} \log_{e} a + \frac{2}{3} \log_{e} b < \log_{e} (\frac{1}{2} a + \frac{2}{3} b) .$

Answer:

Combinatorics

Paper 1

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Combinatorics

Paper 2

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Combinatorics

Paper 3

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Combinatorics

Paper 4

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Combinatorics

Paper 5

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Combinatorics

Paper 6

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Combinatorics

Paper 7

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Combinatorics

Paper 8

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Combinatorics

Paper 9

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Combinatorics

Paper 10

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Combinatorics

Paper 11

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Combinatorics

Paper 12

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Combinatorics

Paper 13

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Combinatorics

Paper 14

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Combinatorics

Paper 15

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Combinatorics

Paper 16

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