A shopkeeper charges Rs. 25 per item for buying 20 or less items. He gives some rebate if items bought are more. If the items bought are 50 or less, then a rebate of Re. 1 per item and for purchase of more than 50 items, rebate of Rs. 2 per item. Find the cost function. What are the points at which this is not continuous?

Introduction

In business mathematics, cost functions help us understand how pricing changes based on the number of units sold or purchased. In this question, we are given a tiered pricing scheme by a shopkeeper. The task is to derive a cost function that accurately models the total cost based on the quantity of items purchased and to determine the points where the cost function is not continuous.

Understanding the Problem

Let the number of items purchased be denoted by x. The pricing strategy given by the shopkeeper is:

For x ≤ 20: Price per item is Rs. 25 (no rebate).
For 20 < x ≤ 50: Rebate of Re. 1 per item, so price = Rs. 24 per item.
For x > 50: Rebate of Rs. 2 per item, so price = Rs. 23 per item.

Step-by-Step: Finding the Cost Function

We can now define the cost function C(x) piecewise as follows:

For x ≤ 20:
C(x) = 25x
For 20 < x ≤ 50:
C(x) = 24x
For x > 50:
C(x) = 23x

Hence, the cost function is:

C(x) = {
  25x,               if x ≤ 20
  24x,               if 20 < x ≤ 50
  23x,               if x > 50
}

Identifying Points of Discontinuity

We must now check whether the function is continuous at x = 20 and x = 50.

At x = 20

From the left: C(20) = 25×20 = 500
From the right: C(20+) = 24×20 = 480

Since 500 ≠ 480, the function is not continuous at x = 20.

At x = 50

From the left: C(50) = 24×50 = 1200
From the right: C(50+) = 23×50 = 1150

Since 1200 ≠ 1150, the function is not continuous at x = 50.

Graphical Representation

If we were to plot the cost function on a graph, we would see sudden jumps (discontinuities) at x = 20 and x = 50, where the pricing structure changes. These points are breakpoints in the function’s domain where continuity does not hold.

Conclusion

The cost function for the shopkeeper’s pricing structure is a piecewise function with different rates for different ranges of items. It is defined as:

C(x) = 25x, for x ≤ 20
C(x) = 24x, for 20 < x ≤ 50
C(x) = 23x, for x > 50

This function is not continuous at two critical points: x = 20 and x = 50, where there is a sudden change in the price per item. This kind of pricing is common in retail and wholesale environments and is modeled using piecewise functions in business mathematics.