Answer:
Maximum attained at point [tex]\left(\dfrac{11}{3},\dfrac{11}{3},\dfrac{11}{3}\right).[/tex]
Minimum attained at point [tex](0,0,0)[/tex]
Step-by-step explanation:
Write f(x,y,z) as
[tex]f(x,y,z)=\dfrac{xyz}{2(xyz)^{1/2}}=\dfrac{\sqrt{xyz}}{2},[/tex]
and let
[tex]g(x)=x+y+z-11.[/tex]
We have to optimize the function f(x,y,z) subject to g(x,y,z)=0. Using Lagrange multipliers, we have to solve the system of equations below:
[tex]\nabla f(x,y,z)=\lambda \nabla g(x,y,z),[/tex]
[tex]g(x,y,z)=0.[/tex]
Or equivalently:
[tex]f_x=\lambda g_x,[/tex]
[tex]f_y=\lambda g_y,[/tex]
[tex]f_z=\lambda g_z,[/tex]
[tex]x+y+z=11.[/tex]
Now we calculate the partial derivatives of f and g:
[tex]f_x=\dfrac{yz}{4\sqrt{xyz}},\ \ f_y=\dfrac{xz}{4\sqrt{xyz}},\ \ f_x=\dfrac{xy}{4\sqrt{xyz}}.[/tex]
[tex]g_x=g_y=g_z=1.[/tex]
Then we have to solve the system of equations
[tex]\begin{cases}\hfil \dfrac{yz}{4\sqrt{xyz}}=\lambda & (1) \\ \hfil \dfrac{xz}{4\sqrt{xyz}}=\lambda & (2) \\ \hfil \dfrac{xy}{4\sqrt{xyz}} =\lambda & (3) \\ x+y+z=1 & (4) \end{cases}[/tex]
From equation (1) and (2) we get by cancelling the common factor [tex]\dfrac{z}{4\sqrt{xyz}}[/tex] that x = y.
Similarly, using (2) and (3) we get that y = z. Therefore, we have that x = y = z, and by equation (4), we obtain that
[tex]x+y+z=3x=11 \Longrightarrow x=\dfrac{11}{3}[/tex]
Since the function f(x,y,z) is non-negative, then [tex]\left(\dfrac{11}{3},\dfrac{11}{3},\dfrac{11}{3}\right)[/tex] is a point where f attains an absolute maximum, and
[tex]f\left(\dfrac{11}{3},\dfrac{11}{3},\dfrac{11}{3}\right)=\dfrac{11\sqrt{33}}{8}\approx 3.51[/tex]
Because of the non-negativity of the function, we see that at [tex](0,0,0)[/tex] f attains an absolute minimum, and its value is
[tex]f(0,0,0)=0.[/tex]
