How to find x values the turning points of a polynomial?

Question

Say as an example I have the polynomial

$$ y = ax^2 + bx^4.$$

How would I find the $x$ values of the turning points?

Answer 1

To find the turning points (stationary points) we look for the stationary points first, and then find the classification of the points.

For your problem $$ y = ax^2 + bx^4 $$ we find the stationary points $$ y' = 2ax + 4bx^3 = 0 = 2(a + 2bx^2)x $$ so we have 3 points of interest

• $x = 0$
• $x = \pm \sqrt{-\frac{a}{2b}}$

To find the classification we take the second derivative $$ y'' = 2a + 12bx^2 $$ for $x=0$ we have $$ y'' = 2a $$ and for $x = \pm \sqrt{-\frac{a}{2b}}$ we have $$ y'' = 2a + 12b\left(-\frac{a}{2b}\right) = 2a -6a = -4a $$ so depending on sign of $a$, we will find out what the points are, for example $ a > 0$ we have

• $x=0 \implies \text{minimum}$
• $x = \pm \sqrt{-\frac{a}{2b}} \implies \text{maximum}$

If $a=0$ then we have inflexion points.