The function f(x, y,z)equals=(xyz )superscript 1 divided by 2(xyz)1/2 has an absolute maximum value and absolute minimum value subject to the constraint xplus+yplus+zequals=11 with xgreater than or equals≥0, ygreater than or equals≥0, and zgreater than or equals≥0. use lagrange multipliers to find these values.

Maximum attained at point

Minimum attained at point

Step-by-step explanation:

Write f(x,y,z) as

and let

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:

Or equivalently:

Now we calculate the partial derivatives of f and g:

Then we have to solve the system of equations

From equation (1) and (2) we get by cancelling the common factor 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

Since the function f(x,y,z) is non-negative, then is a point where f attains an absolute maximum, and

Because of the non-negativity of the function, we see that at f attains an absolute minimum, and its value is