Question

Using Lagrange Multipier, maximize/minimize f(x,y,z)=cos(x)cos(y)cos(z) subject to x+y+z=pi

Answer 1

f = cos(x)cos(y)cos(z)

let g= x+y+z - pi ( from the constraint )

now we require the grad f = scalar multiple of g

let the scalar be Y

ie grad f = Y grad g

Answer 2

grad f = ( -sin(x)cos(y)cos(z) , -cos(x)sin(y)cos(z) , -cos(x)cos(y)sin(z) )

grad g = ( 1,1,1)

Answer 3

therefore

-sin(x)cos(y)cos(z) = Y
-cos(x)sin(y)cos(z) = Y
-cos(x)cos(y)sin(z) = Y