Let `I` be an interval disjointed from `[-1,\ 1]` . Prove that the function `f(x)=x+1/x` is increasing on `I` .

Explanation

`f(x)=x+1/x`

`impliesf'(x)=1-1/x^2`

`f'(x)=0implies1/x^2=1impliesx=+-1`

The points `x=1` and `x=−1` divide the real line into three disjoint intervals i.e., `(-infty.-1],[-1,1] and [1,infty)`

In interval `[−1,1]`, it is observed that:

`x^2le1`

`implies1-1/x^2le0`

Thus f is strictly decreasing on (-1,1) and `f'(x)=1-1/x^2>0 ` on `(-infty,-1) and (1,infty)`

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