Let's sketch the function
f[x_] := 4 x / (x ^ 2 + 3 x + 5)
The derivatives are:
{f'[x], f''[x], f'''[x]} // Together
To get the extreme values of f, compute the zeroes of the first derivatives:
extremes = Solve[f'[x] == 0, x]
And test the second derivative:
f''[x] /. extremes // N
Thus, there is a local maximum at x = Sqrt[5] and a local minimum at x = -Sqrt[5].
Compute the inflection points numerically, chopping imaginary parts close to 0:
inflections = Solve[f''[x] == 0, x] // N // Chop
Insert into the third derivative:
f'''[x] /. inflections
Being different from 0, all three points are actual inflection points.f is not defined where its denominator is 0:
Solve[Denominator[f[x]] == 0, x]
These are non-real numbers, consequently f is defined on all real numbers.
The behaviour of f at the boundaries of its definition:
Limit[f[x], x -> Infinity]
Limit[f[x], x -> -Infinity]
Finally, let's plot f:
Plot[f[x], {x, -8, 6}]