# Global minimum of even-degree polynomial

I'm trying to find the global minimum of a even-degree polynomial (with positive leading coefficient). I could take the derivative of the polynomial, solve that, evaluate original polynomial at each of its roots and take the minimum.

But I was wondering if there is a better way? I'm not interested in all extrema (i.e. the derivative), nor in all minima, just the global one so I was thinking, maybe the task becomes a bit simpler?

• A global minimum only exists for polynomials of even degree $\ge 2$ with positive coefficient for the highest power. And in general, there is no better way to find that point (analytically). You will, however, face problems when trying to (analytically) find the roots of the derivative, if it's degree is $5$ or higher. – Thomas Jun 9 '17 at 10:15
• You can probably do something complex analytic, since all polynomials and their derivatives are which could be useful computationally speaking. But the theory will be more difficult to understand. – mathreadler Jun 9 '17 at 10:23
• @Thomas Yes, I meant even degree. I updated the question, I apologise. – Ecir Hana Jun 9 '17 at 10:29