How to find the maximum and minimum of a function using Maxima?
I have trying to find a way to obtain the maximum and the minimum value of a function using Maxima (wxMaxima), but until now I have not found how to do it.
Could you please tell me how would you do it?
For example, suppose I have the following code:
f(x) := (3*x)/(x^2 - 2*x + 4);
And then I plot this function in the range -10, 10, and I obtain:
I know that the maximum is 3/2 and the minimum should be -1/2.
Answer
My advice is to find the extreme values the same way you would do it by hand: compute the derivative, solve for derivative = 0, and substitute any values found back into the original function. E.g.:
(%i1) f(x) := (3*x)/(x^2 - 2*x + 4);
3 x
(%o1) f(x) := ------------
2
x - 2 x + 4
(%i2) diff (f(x), x);
3 3 x (2 x - 2)
(%o2) ------------ - ---------------
2 2 2
x - 2 x + 4 (x - 2 x + 4)
(%i3) ratsimp (%);
2
3 x - 12
(%o3) - -----------------------------
4 3 2
x - 4 x + 12 x - 16 x + 16
(%i4) num (%);
2
(%o4) 12 - 3 x
(%i5) solve (%, x);
(%o5) [x = - 2, x = 2]
(%i6) map (lambda ([e], subst (e, f(x))), %);
1 3
(%o6) [- -, -]
2 2
If I were being careful, I would have verified that x = -2 and x = 2 are indeed extreme values and not just inflection points, and I would have verified that the denominator of %o3 is nonzero at x = -2 and x = 2 before trying to evaluate f(x) at those points.