If it was a quadratic, you could (in effect) complete the square and thereby get it to a form like

  a(x - b)^2 + c

At that point, you get the answer by inspection. I.e., in this case, if a is positive, then the function is minimized when x = b, and the minimum value is c.

In lucky cases, fourth-order polynomials could be put in such a form. Odd-degree polynomials will not have a unique maximum.