Can roots be at the same points as POI's?

[frenkie]

can roots be at the same points as POI's? because when I set my original equation to zero and I set my second derivative equation to zero i get the same answer? is it possible that roots can be same as the POI's.

 

[quantumdude]

Yes. Just look at $f(x)=x^3$ in any interval containing $x=0$.

 

[robert Ihnot]

Of course, and if you plot Y=X^3, you will see the change of slope at X=0.

 

[quantumdude]

Something else to ponder:

In your precalculus course you learned that you can shift a function vertically by adding a real number to it. That means that no matter what the y-coordinate of the POI of a function, you can always construct another function whose POI is a root, by doing an appropriate vertical shift.