Im having a hard time finding the intervals where the function: f[x_] := x^2 - 4x + 7 Cos[x] is increasing and decreasing using mathematica. I tried using FindRoot but it's not too accurate because the curve has inflection points beyond just the zeroes.
I did this so far, which is wrong. I'm trying to find where it increases and decreases on the interval $[-4,4]$
Taking h[x_] := x^2 - 4x + 7 Cos[x], you can find all roots of h' in the interval $[-4,4]$ with
Solve[h'[x] == 0 && -4 <= x <= 4, x]
However, you can get the intervals where h is increasing and decreasing directly as inequalities with Reduce:
increasing = Reduce[h'[x] > 0 && -4 <= x <= 4, x]
decreasing = Reduce[h'[x] < 0 && -4 <= x <= 4, x]
You can also generalize to outside of the interval $[-4,4]$:
allincreasing = Reduce[h'[x] > 0, x]
alldecreasing = Reduce[h'[x] < 0, x]
If you'd like the numeric forms of the roots, simply apply N, e.g. N @ increasing.
h'' instead of h'—"concavity" is exactly whether the derivative is increasing or decreasing, so you just repeat the procedure with the derivative of the derivative!
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