Demonstrate the inequality a +1/a>=2

To examine the inequality a+1/a >2.

Solution:

Let f(a) =a+1/a >=2. This implies,

(a^2-2a+1)^2/a >=0 or

f(a) = (a-1)^2/a >=0

Now the numerator is positive for all a ,postive or negative and zero when a= 1. But the denominator is positive only for a>0. Therefore, f(a) is positive only for a>0, and equals to zero when a = 1.

Or  f(a) = a+1/a >= 2 for all x>0 and

At a = 0+   f(0+)  = +inf

At a=0- , f(0-) =-inf.

So at a = 0,  f(a) undefined and excluded.

Therefore.

a+1/a >=2 is only valid for a > 0  and does not hold for x < or = 0. or

a+1/a remains undefined for x=0.So,a+1/a >2 does not hold for x = 0.

a+1/a < 2 for x<0. So a+1/a >2 doesnot hold when x<0