Give an example of a polynomial that is concave up on R and whose second derivative has at least two distinct roots.
Lets start with what you want.
To be concave up the 2nd derivative must be greater than zero.
and we want it to have two distinct real roots.
The only way it can have roots and not cross the x-axis is if they are repeated and of even degree.
Then for any real numbers
This satisfies the above now you just need to find f.
This should get you started.