Curve sketching

Let's sketch the function

The derivatives are

To get the extreme values of f, compute the zeroes of the first derivatives:

And test the second derivative:

Thus, there is a local maximum at x = Sqrt[5] and a local minimum at x = -Sqrt[5]. Compute the inflection points numerically, choping imaginary parts close to 0:

Insert into the third derivative:

Being different from 0, all three points are actual inflection points. f is not defined where its denominator is 0:

These are non-real numbers, consequently f is defined on all real numbers. The behaviour of f at the boundaries of its definition:

Finally, let's plot f: