1. ## concave polynomials

Give an example of a polynomial that is concave up on R and whose second derivative has at least two distinct roots.

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.