Consider the third degree polynomial equation as $\"\"$ .

Where $\"\"$ and $\"\"$ are the roots of the polynomial equation.

Apply derivative on each side with respect to $\"\"$ .

$\"\"$ .

Product rule of derivative : $\"\"$ .

$\"\"$

Again apply derivative on each side with respect to $\"\"$ .

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

Therefore the inflection point is at $\"\"$ .

The $\"\"$ -coordinate of the inflection point is $\"\"$ .

The third degree polynomial equation has only one inflection point .

The $\"\"$ -coordinate of the inflection point is $\"\"$ .