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Obtain the general solution for this equation?

Posted 4 months ago
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I was working on this problem as a demonstration for Methods

g[x_] = Sin[x]/E^x;
Reduce[g'[x] == 0, x] produces 
C[1] \[Element]   Integers && (x == -2 ArcTan[1 + Sqrt[2]] + 2 \[Pi] C[1] || 
   x == -2 ArcTan[1 - Sqrt[2]] + 2 \[Pi] C[1])

Solve[g'[x] == 0, x] produces the first solutions between -Pi and Pi {{x -> -((3 [Pi])/4)}, {x -> [Pi]/4}}

Solve[{g'[x] == 0, 0 <= x <= 4 Pi}, x] produces something similar to the reduce answer {{x -> 2 \[Pi] - 
2 ArcTan[1 - Sqrt[2]]}, {x -> -2 ArcTan[1 - Sqrt[2]]}, {x -> 
2 \[Pi] - 2 ArcTan[1 + Sqrt[2]]}, {x -> 
4 \[Pi] - 2 ArcTan[1 + Sqrt[2]]}}

I would like to know why it isn't producing the simpler solution x=(kPi+1)/4 k element of integers

Also I wondered why Mathematica joins the graph of h'[x] at 2 for the hybrid function

h[x_] = Piecewise[{{-4, x <= 0}, {x^2 - 4, 0 < x < 2}, {x - 2, 
x >= 2}}]; 

when it is not continuous.

For the last question, this does the trick:

Plot[PiecewiseExpand[h'[x]], {x, -2, 3}]
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