Let f (x, y) = xy + 5x + 4y model the utility a consumer feels based on the number of the

number of cups of coffee, x, and the number of donuts, y, consumed on a given Thursday. Find
the absolute maximum and absolute minimum values of f(x, y) over the triangular region with
vertices (0,0), (2,0), and (0,2) . Be sure to include any critical points found on each boundary as
well as any critical points inside the region. Hint: The equation of the diagonal boundary
is y = −x + 2 .

To find the absolute maximum and minimum values of f(x, y) = xy + 5x + 4y over the given triangular region, including critical points on each boundary and inside the region, we can follow these steps:

Step 1: Identify the boundaries of the triangular region:
The triangular region is defined by the vertices (0,0), (2,0), and (0,2). The diagonal boundary is defined by the equation y = -x + 2.

Step 2: Find critical points in the interior of the region:
To find critical points inside the region, we need to find where the partial derivatives of f(x, y) with respect to x and y equal zero.
The partial derivatives are:
∂f/∂x = y + 5
∂f/∂y = x + 4

Setting both partial derivatives equal to zero:
y + 5 = 0 --> y = -5
x + 4 = 0 --> x = -4

Therefore, the critical point inside the region is (-4, -5).

Step 3: Find critical points on the boundary:
Now we need to evaluate f(x, y) at the three vertices and on the diagonal boundary.

At (0,0):
f(0, 0) = (0)(0) + 5(0) + 4(0) = 0

At (2,0):
f(2, 0) = (2)(0) + 5(2) + 4(0) = 10

At (0,2):
f(0, 2) = (0)(2) + 5(0) + 4(2) = 8

Along the diagonal boundary (y = -x + 2), substitute that expression for y in f(x, y):

f(x, -x + 2) = x(-x + 2) + 5x + 4(-x + 2)
= -x^2 + 2x + 5x - 4x + 8
= -x^2 + 3x + 8

Step 4: Determine the absolute maximum and minimum:
We have evaluated f(x, y) at all the critical points on the boundary and the critical point inside the region. Now we compare their values to find the absolute maximum and minimum:

At critical point (-4, -5):
f(-4, -5) = (-4)(-5) + 5(-4) + 4(-5)
= 20 - 20 - 20
= -20

At (0,0):
f(0,0) = 0 (previously calculated)

At (2,0):
f(2,0) = 10 (previously calculated)

At (0,2):
f(0,2) = 8 (previously calculated)

Now let's find the maximum and minimum values along the diagonal boundary:

f(x, -x + 2) = -x^2 + 3x + 8

To find the maximum and minimum values on this boundary, we need to find where the derivative of f(x, -x + 2) with respect to x equals zero.
So, differentiate f(x, -x + 2) with respect to x:

df/dx = -2x + 3

Setting df/dx equal to zero and solving for x:
-2x + 3 = 0
-2x = -3
x = 3/2

Substitute this value of x back into f(x, -x + 2) to find the corresponding y value:

f(3/2, -3/2 + 2) = f(3/2, 1/2) = (3/2)(1/2) + 5(3/2) + 4(1/2)
= 3/4 + 15/2 + 2
= 3/4 + 30/4 + 8/4
= 41/4

So, the maximum value along the diagonal boundary is 41/4.

Step 5: Finalize the results:
Comparing all the values obtained, we have:

Absolute maximum: f(3/2, 1/2) = 41/4
Absolute minimum: f(-4, -5) = -20

Therefore, the absolute maximum value of f(x, y) = xy + 5x + 4y over the triangular region is 41/4, and the absolute minimum value is -20.