First, a few simple observations:
- must be a polynomial of even degree
- Taking derivatives enough times will eventually give zero
- The function is a finite sum, and all of the derivatives are polynomials, so is a polynomial
- Since has even degree, has even degree too
Now, to show that , we should just try to find its minimum. (And in this case, it is enough to do so. Be sure to justify this!) To do this, we set the derivative equal to zero:
and "solve". Suppose is a solution. We can put it back into to see what value we get:
Since is positive for all , we have that . Therefore, all minimum values of are greater than or equal to zero, so all of must be greater than or equal to zero. This proves it!