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How demonstrate ln x> or equal 2(x*-1)/(x+1) if x>1?

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sparangello | Student, College Freshman | eNoter

Posted August 19, 2012 at 1:28 PM via web

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How demonstrate ln x> or equal 2(x*-1)/(x+1) if x>1?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 19, 2012 at 1:57 PM (Answer #1)

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You should for the following function such that:

`f(x) = ln x - 2(x-1)/(x+1) `

The problem requests to prove that `f(x) >= 0 < = > x > 1`

You need to prove that the function reaches its minimum at x = 1, hence, you need to evaluate the zeroes of the equation `f'(x) = 0`  such that:

`f'(x) = 1/x - (2(x+1)-2(x-1))/((x+1)^2)`

`f'(x) = 1/x - (2x+2-2x+2)/((x+1)^2)`

`f'(x) = 1/x - 4/((x+1)^2) => f'(x) = (x^2 + 2x + 1 - 4x)/(x(x+1)^2)`

`f'(x) = (x^2 -2x + 1)/(x(x+1)^2) =gt f'(x) = ((x-1)^2)/(x(x+1)^2) `

You need to solve `f'(x) = 0`  such that:

`((x-1)^2)/(x(x+1)^2) = 0 =gt (x-1)^2 = 0 =gt x_1 = x_2 = 1`

Hence, since the zeroes of derivative are `x_1 = x_2 = 1` , then the  function reaches its extreme at `x = 1` .

Notice that derivative is positive for `xgt0`  and it is negative for `xlt0` , hence, if `x in (1,oo),`  the function `f(x) = ln x - 2(x-1)/(x+1)`  is positive, thus `ln x - 2(x-1)/(x+1) gt= 0 =gt ln xgt= 2(x-1)/(x+1).`

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