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

1. Mar 28, 2006

### 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.

2. Mar 28, 2006

### Tom Mattson

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

3. Mar 29, 2006

### robert Ihnot

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

4. Mar 29, 2006

### Tom Mattson

Staff Emeritus
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.