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.
$\displaystyle f''(x) \ge 0$ 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 $\displaystyle r_1 \ne r_2$
$\displaystyle f''(x)=(x-r_1)^2(x-r_2)^2$
This satisfies the above now you just need to find f.
This should get you started.