Answer
$f\left( {1,1,1} \right) = 1$
Work Step by Step
$$\eqalign{
& {\text{Maximize }}f\left( {x,y,z} \right) = xyz \cr
& {\text{Constraint: }}x + y + z - 3 = 0{\text{ }} \to {\text{ }}g\left( x \right) = x + y + z \cr
& \cr
& {\text{We need to maximize }}f\left( x,y,z \right) = xyz{\text{ }}\left( {{\text{objective function}}} \right) \cr
& {\text{Using the Method of Lagrange Multipliers}} \cr
& *{\text{ }}{f_x}\left( {x,y,z} \right) = \lambda {g_x}\left( {x,y,z} \right) \cr
& \frac{\partial }{{\partial x}}\left[ {xyz} \right] = \lambda \frac{\partial }{{\partial x}}\left[ {x + y + z} \right] \cr
& yz = \lambda \left( 1 \right) \cr
& yz = \lambda {\text{ }}\left( {\bf{1}} \right) \cr
& \cr
& *{\text{ }}{f_y}\left( {x,y,z} \right) = \lambda {g_y}\left( {x,y,z} \right) \cr
& \frac{\partial }{{\partial y}}\left[ {xyz} \right] = \lambda \frac{\partial }{{\partial y}}\left[ {x + y + z} \right] \cr
& xz = \lambda \left( 1 \right) \cr
& xz = \lambda {\text{ }}\left( {\bf{2}} \right) \cr
& \cr
& *{\text{ }}{f_z}\left( {x,y,z} \right) = \lambda {g_z}\left( {x,y,z} \right) \cr
& \frac{\partial }{{\partial z}}\left[ {xyz} \right] = \lambda \frac{\partial }{{\partial z}}\left[ {x + y + z} \right] \cr
& xy = \lambda \left( 1 \right) \cr
& xy = \lambda {\text{ }}\left( {\bf{3}} \right) \cr
& \cr
& {\text{Solving the system of equations }}\left( {\bf{1}} \right){\text{, }}\left( {\bf{2}} \right){\text{ and }}\left( {\bf{3}} \right) \cr
& {\text{Equating the equations }}\left( {\bf{1}} \right){\text{ and }}\left( {\bf{2}} \right) \cr
& yz = \lambda ,{\text{ }}xz = \lambda \cr
& yz = xz \cr
& y = x \cr
& {\text{Equating the equations }}\left( {\bf{1}} \right){\text{ and }}\left( {\bf{3}} \right) \cr
& yz = \lambda ,{\text{ }}xy = \lambda \cr
& yz = xy \cr
& z = x \cr
& \cr
& {\text{Substituting }}y = x{\text{ and }}z = x{\text{ into the constraint }} \cr
& x + y + z - 3 = 0 \cr
& x + x + x - 3 = 0 \cr
& 3x = 3 \cr
& x = 1 \cr
& {\text{Then}}{\text{, }}y = 1{\text{ and }}z = 1 \cr
& \cr
& {\text{Therefore}}{\text{,}} \cr
& f\left( {1,1,1} \right) = \left( 1 \right)\left( 1 \right)\left( 1 \right) \cr
& f\left( {1,1,1} \right) = 1 \cr
& \cr
& {\text{The maximum is }}1{\text{ at }}x = y = z = 1 \cr} $$
