Need help with the following questions? (Serious answers please).?

Question

1. The cost of producing x thousands of a certain product is given by:
C(x) = 9x - 3x^2 + 2x^2
At what production level x is the marginal cost Increasing and decreasing?

2. Find the critical points for the following functions and determine whether the function has a local maximum or local minimum at each critical point.
(a). f(x) = xe^(-x)
(b). f(x) = 6+x -x^2






























. The cost of producing x thousands of a certain product is given by:
C (x) = 9x -3x^2 + 2x^3

At which production level x is the marginal cost:
Increasing and decreasing?

2. Find the critical points for the following functions and determine whether the function has a local maximum or local minimum at each critical point.
(a). f(x) - xe^(-x).
(b). f(x) = 6 + x-x^2

Answer 1

1. First: -3x^2 + 2x^2 = x^2 so C = 9x - x^2
dC/dx = 9 - 2x = 0 and x=4.5
d2C/d2x = -2x = -9 at x=4.5
So this is the maximum of the function

2. (a) df/dx = e^(-x) -xe^(-x) = 0 => 1 - x = 0 and x = 1
d2f/d2f = -e^(-x) - e^(-x) + xe^(-x)
evaluate at 1: -e^(-1) - e^(-1) + e^(-1) = -e(-1)
Since the second derivative is <0 x=1 is a maximum
(b) df/dx = x - 2x and x=1/2
d2f/d2x = -2x = -1 at x=1/2 so this is a maximum