# Curve sketching

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, choping 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}]