Question

Find the maximum value if sin x*sin y*sin(x +y)

Answer 1

Calling the expression mm and focussing on just one period,

∂m∂x=cosxsinysin(x+y)+sinxsinycos(x+y)=0∂m∂x=cos⁡xsin⁡ysin⁡(x+y)+sin⁡xsin⁡ycos⁡(x+y)=0

and

∂m∂y=sinxcosysin(x+y)+sinxsinycos(x+y)=0∂m∂y=sin⁡xcos⁡ysin⁡(x+y)+sin⁡xsin⁡ycos⁡(x+y)=0

from which

cosxsinysin(x+y)=sinxcosysin(x+y)⇒sin(x+y)sin(x−y)=0cos⁡xsin⁡ysin⁡(x+y)=sin⁡xcos⁡ysin⁡(x+y)⇒sin⁡(x+y)sin⁡(x−y)=0.

Then x=yx=y. From this sinx=0sin⁡x=0 or \sin 3x=0 the latter giving x=π3x=π3 giving a maximum of 33√8338.

I leave it to you to determine the minimum and track through the 2 ignored situations.