Show that is increasing in R.

Given:

To show: the given function is increasing in R.

Explanation: Given

Applying first derivative with respect to x, we get

But the derivative of 2x is 2, so

But the derivative of , so

Applying the sum rule to last part we get

cancelling the like terms, we get

Now adding by taking the LCM, we get

Now for any real value of x, the above value of f(x) is greater than or equal to zero

Hence

And we know, if f’(x)≥0, then f(x) is increasing function.

Hence the given function is increasing function in R.