The demand for a certain commodity is d(x) = 5000e−.02x units per month when the market price is x dollars per unit.

(a) at what rate is the consumer expenditure e(x) = xd(x) changing with respect to price x when the price is equal to $110 dollars?
(b) at what price does consumer expenditure stop increasing and begin to decrease?
(c) at what price does the rate of consumer expenditure begin to increase?

Step-by-step explanation:

We have two functions:

firstly, we can express D(x) in the above equation.

Thinking of the questions mathematically,

part a) is only asking what is the rate of change of the expenditure(E) with respect to x(price) when x is 110 dollars? that is,

evaluated at x = 110.

using the product rule for differentiation.

for simplification we can take e^0.02x common

now put x= 110, and we'll get the rate of change of the expenditure at x = 110. (answer for part a):

part(b) at what value of x does E'(x) = 0? i.e, at what price(x) does the expenditure E changes from increasing to decreasing?

we can split the equation and solve them separately.

at x =50 dollars, the price changes from increasing to decreasing.

part(c) is only asking: from what value of x is the price increasing.

Well, judging from the answer of E'(x) = 0, the only value from where the price is changing from increasing to decreasing is x =50. so, the price has been increasing from x < 50. BUT since price cannot be negative (depending on the context), we can't say , instead our answer should be: that the rate of consumer expenditure (dE/dx) begin increasing from x = 0.