The profit generated by an information technology consulting...

The profit generated by an information technology consulting firm can be modeled as $P(x, y) = -4x^2 + 96x + 168y - 7y^2$ millon dollars where x thousand hours are logged by on-site desktop engineers and y thousand hours are logged by network systems engineers. (a) Calculate the point of maximized profit. x = y = thousand hours thousand hours (b) Verify that the result of part (a) is a maximum point. Find the second partials. $P_{xx} = $ $P_{yy} = $ $P_{xy} = $ $P_{yx} = $ Using the Determinant Test, $D(x, y) = $ Since D ? 0 and $P_{xx}$ ? 0, the critical point is a relative maximum.

Transcript

00:01 The first step to find critical points of any function is to take the first derivative.

00:05 In this case, since we have two variables in the function, we have x and we have y, we need to take the partial derivatives.

00:12 So first, the derivative with the derivative of p with respect to x, the partial derivative, to be precise.

00:21 That means 560, it has an x, so it's just 560.

00:27 This one we treated as a, y and 20 a constant, so 20y.

00:34 And this would be minus 40x.

00:38 The last one cancels because it doesn't have any x.

00:42 And we also need to take the partial derivative of p with respect to y.

00:48 So this is a constant.

00:50 We don't write it.

00:51 This is 20x.

00:54 This one doesn't have any y's.

00:56 So treat us a constant.

00:58 And this one would be minus 3.

01:00 Y minus 12 sorry six times two is 12 12.

01:07 Now we set each one of these derivatives equal to 0 and let's open some space here.

01:14 Now would we have enough space to do some calculations? let's do the top one.

01:19 If we solve it equal to 0, we can, let's set it in terms of x.

01:29 So 560 plus plus 20y is equal to 40x.

01:38 And now we divide all of them by 40 and write the simplification version of each of the fractions because this is going to be 560 by 40, 20 by 40 and 40 by 40.

01:53 So x is going to be equal to 560 divided by 40 is equal to 14.

02:01 And 20 divided by 40 is the same as 1 half, so y over 2.

02:07 And we have an equation of x in terms of y.

02:10 And then we replace this in this equation.

02:14 And instead of writing 20x, we write 20 times 14 plus y over 2 minus 12 y equals 0.

02:25 And here we can solve for y to find the y value is equal to 140.

02:37 Then we plug it back into this equation, so 14 plus 140 divided by 2, to get that x is equal to 84...