Let \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}} + \frac{1}{{\rm{x}}}\), where x ∈ (0, 1). Then which one of the following is correct?

Concept:

Monotonic function: If a function is differentiable on the interval (a, b) and if the function is increasing/strictly increasing or decreasing/strictly decreasing, then the function is known as monotonic function.

First derivative test:

• If \(\frac{{{\rm{df}}\left( {\rm{x}} \right)}}{{{\rm{dx}}}} \ge 0\) for all x in (a, b) then, f(x) is an increasing function in (a, b).
• If \(\frac{{{\rm{df}}\left( {\rm{x}} \right)}}{{{\rm{dx}}}} \le 0\) for all x in (a, b) then, f(x) is an decreasing function in (a, b).

For strictly increasing or decreasing:

• \(\frac{{{\rm{df}}\left( {\rm{x}} \right)}}{{{\rm{dx}}}} > 0\) for strictly increasing.
• \(\frac{{{\rm{df}}\left( {\rm{x}} \right)}}{{{\rm{dx}}}} < 0\) for strictly decreasing.

Calculation:

Given: \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}} + \frac{1}{{\rm{x}}}\)

Differentiating w.r.t x

\(\Rightarrow {\rm{f'}}\left( {\rm{x}} \right) = 1 - \frac{1}{{{{\rm{x}}^2}}}\)

\(\Rightarrow {\rm{f'}}\left( {\rm{x}} \right) =\rm \frac{x^2 -1}{x^2}= \rm \frac{(x -1)(x+1)}{x^2}\)

So, for x ∈ (0, 1), f’(x) < 0.

Hence, f(x) is decreases for x ∈ (0, 1).