There is a function $x^3 – 6x^2 + 9x + 1$.
Its critical points are $1$ and $3$.
I am very confused, if these points are maximum and minimum points respectively or are both inflection points. Can someone please help me with these ?
These points can not be maximum and minimum points since function attains higher and lower values as compared to what the function attains at these $2$ points. Please correct me if I am wrong.
Best Answer
Indeed, a degree three polynomial has never maximum or minimum as it tends to plus and minus infinity. Yet, what is usually asked for in such contexts are local maximum and minimum.
A way to find out if you have those is to consider the second derivative at those points. If it is negative it is a local maximum, if it is positive it is a local minimum and if it is zero it is an inflection point.