Question

What are the absolute extrema of `f(x)=(sinx) / (xe^x) in[ln5,ln30]`?

Answer 1

`x = ln(5)` and `x = ln(30)`

Explanation:

I guess the absolute extrema is the "biggest" one (smallest min or biggest max).

You need `f'` : `f'(x) = (xcos(x)e^x - sin(x)(e^x + xe^x))/(xe^x)^2`

`f'(x) = (xcos(x) - sin(x)(1 + x))/(x^2e^x)`

`AAx in [ln(5),ln(30)], x^2e^x > 0` so we need `sign(xcos(x) - sin(x)(1 + x))` in order to have the variations of `f`.

`AAx in [ln(5),ln(30)], f'(x) < 0` so `f` is constantly decreasing on `[ln(5),ln(30)]`. It means that its extremas are at `ln(5)` & `ln(30)`.

Its max is `f(ln(5)) = sin(ln(5))/(ln(25))` and its min is `f(ln(30)) = sin(ln(30))/(30ln(30))`