Answer:
Explained below.
Step-by-step explanation:
(a)
It is provided that the price function for 1000 TVs is,
<em>p </em>(1000) = 400.
Also provided that if rebate of $10 is given then sale increases by 100 per week.
Let <em>x</em> be the number of unit sold per week then (<em>x</em> − 1000) is the increase in the number of units sold.
Then the price function is:
Thus, the demand function is, .
(b)
The revenue function is:
Maximize the revenue as follows:
Observe that R'(x) > 0 for 0 ≤ x < 2500 and R'(x) < 0 for x > 2500. Hence, first derivative test will lead to the conclusion that maximum occurs at x = 2500.
Compute the value p (2500) as follows:
Then the rebate to maximize the revenue should be: $400 - $250 = $150.
(c)
The weekly cost function is,
Compute the profit function as follows:
Compute the marginal profit as follows:
Compute the value of <em>x</em> for P'(x) = 0 as follows:
Observe that P'(x) > 0 for 0 ≤ x < 1950 and P'(x) < 0 for x > 1950. Hence, first
derivative test will lead to the conclusion that maximum occurs at x = 1950.
Compute the value p (1950) as follows:
Then the rebate to maximize the profit should be: $400 - $305 = $95.