Thread: the minimum value

Find the minimum value of the function

.

This problem only works for even degree exponents, it is hard to state the details in general so I will do it for special cases to show how to proceede. Let . So you want to know .

Begin by considering the general problem for and treating your case as a special case. First, start of easy, we want to minimize . Calculus tells us that the minimum will occur at or when does not exist. Thus, we can look for when or when but . Now, since we can look when and that happens when .

By induction we can prove that for even by re-introducting the previous case into the problem.

The above illustrates that we can now only consider the case . Say then . It should seem clear that is a zero by symettry of the coefficients, thus is a factor. By division we get (we can also reach these results by factoring the same coefficient terms but that is too long to type). By division we get . Thus, there are no other real zeros. Which means is the minimum point so which is the minimum value.
(So to complete this proof we need to show has no other zero except for , that should be doable by using the lower bound estimate to get which was specifically demonstrated for ).