Question

Determine the global extreme values of the f(x,y)=9x−4y if y≥x−5,y≥−x−5,y≤7. (Use symbolic notation and fractions where needed.) Incorrect

Answer 1

To find the global extreme values of the function f(x, y) = 9x - 4y, we solve the system of inequalities to find the critical points, and then evaluate the function at the critical points and at the endpoints of the feasible region. The global maximum value is 65 and the global minimum value is -55.

Step-by-step explanation:

To find the global extreme values of the function f(x, y) = 9x - 4y, we need to determine the values of x and y that will give us the maximum and minimum value of the function. To do this, we need to find the critical points and evaluate the function at these points and at the endpoints of the feasible region.

Step 1: Find the critical points by solving the system of inequalities.

Step 2: Evaluate the function at the critical points and at the endpoints of the feasible region.

Step 3: Compare the values obtained in step 2 to find the global extreme values.

In this case, the critical points are (x, y) = (4, -1) and (x, y) = (3, 0). The function is evaluated as f(4, -1) = 56 and f(3, 0) = 57. The endpoints of the feasible region are (x, y) = (5, 0) and (x, y) = (2, 7). Evaluating the function at these endpoints gives f(5, 0) = 65 and f(2, 7) = -55.

Therefore, the global maximum value is 65 and the global minimum value is -55.

Answer 2

To determine the global extreme values of the function f(x,y) = 9x - 4y subject to the given conditions, we find the critical points and evaluate the function at the boundary. The global minimum value is -25 and the global maximum value is 35.

Step-by-step explanation:

To determine the global extreme values of the function f(x,y) = 9x - 4y subject to the conditions y ≥ x - 5, y ≥ -x - 5, and y ≤ 7, we first need to find the critical points of the function within the given region. We do this by finding the points where the derivative of the function is equal to zero. Once we have the critical points, we evaluate the function at these points as well as at the boundary of the region to find the extreme values.

Step 1: Find the critical points by taking the partial derivatives of the function with respect to x and y:
fx = 9 and fy = -4. Equating these to zero, we find that there are no critical points since the partial derivatives are constants.

Step 2: Evaluate the function at the boundary:
Substituting y = x - 5 into the function, we get f(x,x-5) = 9x - 4(x - 5) = 5x + 20. Evaluating this function at the endpoints of the y ≤ 7 condition, we find that the minimum value is obtained at (x,y) = (0, -5) with a value of -25, and the maximum value is obtained at (x,y) = (0, 7) with a value of 35.

Therefore, the global minimum value of f(x,y) is -25 and the global maximum value is 35.