We want find where x → 21*x^4 + 73*x^3 - 57*x^2 + 73*x - 78 (polynomial of even degree) takes the minimum value by using min0.htm (for the maximun value I can use max0.htm). Suppose we want to know the rounded value to 6 digits. We prove with a = -10 and b = 10, and click "test".
For x → ∞ and for x → −∞ a polynomial function of degree 4 tends to ∞. I understand that min is between -6 (8880) and 2 (760). If I used a program that allows me to draw graphs, for example this script (or R or Desmos), I would get:
Anyway, starting from a = -6, b = 2, I get:
The minimum value is -1087.8428466719... reached when x = -3.1295325..., rounding: -3.12953.
I cannot take many figures as the calculation is affected by the approximations in the tabulation of the function, and it would make no sense, in any practical problem, to take many digits.
If (for some reason) I needed more digits, I would have to resort to looking for where the first derivative is 0 with a program for the solution of polynomial equations. I find: -3.12953249066755.