Final answer:
The absolute maximum value of the function is 2ε at (x,y)=(6,0) and the absolute minimum value of the function is -1 at (x,y)=(-6,0).
Step-by-step explanation:
The given function is z=f(x,y)=39x+12y+50. The constraint given is y≥0 and y≤36−x. This is a closed and bounded feasibility region, so we are guaranteed to have both an absolute maximum and an absolute minimum value of the function. The absolute maximum and minimum values can be found by using the Lagrange multipliers method.
First, we need to find the critical points of the function. To do this, we need to take the partial derivatives of the function with respect to x and y. We get:
∂f/∂x=39
∂f/∂y=12
We set these equal to zero and solve for x and y, which gives us x=0 and y=0. We then check to make sure these points are within the feasible region. Since x=0 and y=0 satisfy our constraints, we know that these are our critical points.
Next, we need to find the extreme values of the function. To do this, we need to use the Lagrange multipliers method. We need to set up a system of equations with the given constraints and the partial derivatives of the function. We get:
39x+12y+50=λ(1)
y≥0 (2)
y≤36−x (3)
We solve this system of equations by substituting the value of y from constraint (3) into constraint (2). This gives us:
39x+12(36−x)+50=λ (4)
Solving for x, we get x=6. Substituting this value into constraint (3) gives us y=0. Now we can substitute these values into equation (4) to get λ=2ε. This is the absolute maximum value of the function.
To find the absolute minimum value, we need to take the partial derivatives of the function with respect to x and y and set them equal to zero. We get:
∂f/∂x=39
∂f/∂y=12
We set these equal to zero and solve for x and y, which gives us x=-6 and y=0. We then check to make sure these points are within the feasible region. Since x=-6 and y=0 satisfy our constraints, we know that these are our critical points. Substituting these values into the original function gives us an absolute minimum value of -1.
Therefore, the absolute maximum value of the function is 2ε at (x,y)=(6,0) and the absolute minimum value of the function is -1 at (x,y)=(-6,0).