Use calculus. To find min(f(x)) then solve 
. The roots of that give you the critical points; call those 
for the moment. Then, examine 
. Positive values indicate that the corresponding 
was a minima.
. The roots of that give you the critical points; call those for the moment. Then, examine . Positive values indicate that the corresponding was a minima. For a degree 4 polynomial, the derivative is degree 3, and there are closed form solutions for the roots of a polynomial of degree 3. Therefore you can calculate the exact critical points, and test them.
You will only find one real-valued root. You might assume that it must be the minima, but in the general case, it will not be a safe assumption.
