A manufacturer of electric kettles faces the following economic data:
Demand (price–quantity) relation: p=1200−5x,
Total cost function: C(x)=100000+200x+0.2x2,
Indirect tax: An excise duty of Rs. 50 is payable per kettle produced.
Here x (a positive integer) is the number of kettles produced and sold and p (in rupees) is the selling price per kettle.
Respond to the questions below.
a. (i) Formulate the profit function P(x) after including the tax liability.
b. (ii) Determine the production level x and the corresponding selling price p that maximise the manufacturer’s profit.
c. (iii) Calculate the maximum profit that the manufacturer can earn under the given conditions.
d. (iv) Confirm, using an appropriate derivative test, that the value of x found in part (ii) gives a maximum profit.