Prove 1/2 + (1)/(2(u+1)^2) - 1/(sqrt(1+2u)) >= 0 for u>=0

ndevunidt 2022-10-17 Answered
Prove 1 2 + 1 2 ( u + 1 ) 2 1 1 + 2 u 0 for u 0
This inequality provides a tight lower bound to 1 + 2 u for u 0 without a radical. I was trying to solve it by squaring the radical and cross-multiplying and repeated differentiation of the resulting expression, I wonder if there is a quicker solution. Thanks.
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Answers (2)

wlanauee
Answered 2022-10-18 Author has 17 answers
It's
2 u + 1 ( u 2 + 2 u + 2 ) 2 ( u + 1 ) 2
or
( 2 u + 1 ) ( u 2 + 2 u + 2 ) 2 4 ( u + 1 ) 4
or
u 3 ( 2 u 2 + 5 u + 4 ) 0.
Done!
Also, we can use the following way.
Let 2 u + 1 = x
Thus, x 1 and we need to prove that
1 2 + 1 2 ( x 2 1 2 + 1 ) 2 1 x
or
1 + 4 ( x 2 + 1 ) 2 2 x
or
x ( x 2 + 1 ) 2 + 4 x 2 ( x 2 + 1 ) 2
or
x 5 + 2 x 3 + x + 4 x 2 x 4 + 4 x 2 + 2
or
x 5 2 x 4 + 2 x 3 4 x 2 + 5 x 2 0
or
x 5 2 x 4 + x 3 + x 3 2 x 2 + x 2 x 2 + 4 x 2 0
or
( x 1 ) 2 ( x 3 + x 2 ) 0 ,
which is obvious for x 1.
Done againe!
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Gisselle Hodges
Answered 2022-10-19 Author has 5 answers
write your inequality in the form
1 2 + 1 2 ( u + 1 ) 2 1 1 + 2 u
and square it after this we get
1 / 4 u 3 ( 5 u + 2 u 2 + 4 ) ( u + 1 ) 4 ( 1 + 2 u ) 0
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