Unable to formulate the correct formulation for maximize a function

Question

So, i want to express these:

$$\max(\cos(x)\cos(y)\cos(z)),\quad x+y+z=\pi,\quad x,y,z\in(0,\pi]$$

Attempt:

In[81]:= NMaximize[{Cos[x] Cos[y] Cos[z], 
  x + y + z == Pi && 0 < x <= \[Pi] && 0 < y <= \[Pi] && 
   0 < z <= \[Pi]}, {x, y, z}]

Out[81]= {0.125, {x -> 1.0472, y -> 1.0472, z -> 1.0472}}

But how so? I tried manually like this:

In[82]:= N[Cos[30 Degree] Cos[30 Degree] Cos[30 Degree]]

Out[82]= 0.649519

The above calculation implies there are numbers greater than the maximum value, i mean $0.649>0.125$. I think, i made a mistake to formulate the expression. Could you help me please? If the formulation is correct that would be greater or equal to Out[82]. Thanks in advance?

Answer 1

You can use Maximize or ArgMax or MaxValue to confirm your result.

Maximize[{Cos[x] Cos[y] Cos[z], 
  x + y + z == Pi && 0 < x <= π && 0 < y <= π && 
   0 < z <= π}, {x, y, z}]

{1/8, {x -> π/3, y -> π/3, z -> π/3}}

ArgMax[{Cos[x] Cos[y] Cos[z], 
  x + y + z == Pi && 0 < x <= π && 0 < y <= π && 
   0 < z <= π}, {x, y, z}]

{π/3, π/3, π/3}

MaxValue[{Cos[x] Cos[y] Cos[z], 
  x + y + z == Pi && 0 < x <= π && 0 < y <= π && 
   0 < z <= π}, {x, y, z}]

1/8