Help with Inflexion points of a function I have this function: $P(x) = x^4 +cx^3 + \frac{x^2}{24}$ and i need to find for which values of c the function has: a) two inflection points b) one inflection point c) does not have any inflection point I already know that $P''(x) = 12x^2 + 6cx + \frac{1}{12}$ But i don't what to do after that for find the values of c. I would appreciate if somebody can help me.

The second derivative is $P''(x)=12x^2+6cx+\frac{1}{12}$.

$P''(x)=0$ will give the $x$-coordinates of the inflection points.

If the equation $P''(x)=0$ has two solutions, there are two inflection points, if it has one, there is one inflection point, if it has none, there is no inflection point.

The number of solutions to $P''(x)=0$ can be determined using the discriminant, which is $D=b^2-4ac=36c^2-4$.

For $36c^2-4>0$, $P''(x)=0$ has two solutions. For $36c^2-4=0$, $P''(x)=0$ has one solution, for $36c^2-4<0$, $P''(x)=0$ has no solutions.

> Two solutions for $c<-\frac13 \vee c>\frac13$. One solution for $c=-\frac13 \vee c=\frac13$. No solutions for $-\frac13