# How many critical points can a quadratic polynomial function have?

Jul 27, 2018

A quadratic polynomial function can have a single critical point.

#### Explanation:

A quadratic polynomial is a polynomial of second degree, in the form:

$f \left(x\right) = a {x}^{2} + b x + c$

with $a \ne 0$.

By definition the critical points for $f \left(x\right)$ are the roots of the equation:

$\frac{\mathrm{df}}{\mathrm{dx}} = 0$

so:

$2 a x + b = 0$

As this is a first degree equation, it has a single solution:

$\overline{x} = - \frac{b}{2 a}$

so a quadratic polynomial function can have a single critical point, which by the way is the vertex of the parabola of equation:

$y = f \left(x\right)$