Minimum value of the function

Question:

Expression !!{mx - 1 + 1/x}!! is non-negative for all positive real !!x!!.Minimum value of !!m!!?
a)!!0!!
b)!!(-1/2)!!
c)!!(1/2)!!
d)!!(1/4)!!

Solution

Since we want the value of the function to be non negative.

You need to model this into some theorem which relates inequality with the value of function.

Some of us said we can convert this to a quadratic and then look at when does the value of quadratic be non-negative.

Secondly some looked at the AM, GM concept to solve it.

Approach 1:

By applying the dscriminant property of the equation !!{mx^2 - x + 1}!!
For the expression to be non-negative , !!D!! should be !!≤!! !!0!!
That is,
!!(1-4m)!! !!≤!! !!0!! !!m!! !!≤!! !!(1/4)!!

Approach 2:

By applying A.M.,G.M. concept;
Let !!mx!! and !!1/x!! be two no.s such that
!!(mx+1/x)/2!! !!≥!! !!Sqrt(m)!!

That is,
!!(mx + 1/x -2sqrt(m))!! !!≥!! !!0!! Comapring it with the original equation , we get
!!2sqrt(m)!! !!=!! !!1!!

!!m!! !!=!! !!(1/4)!!

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