Problems on profit maximization and
minimization of cost function:
Example 6.27
For a particular process, the
cost function is given by C = 56 - 8x + x2
, where C is cost per unit and x, the number of unit’s produced. Find
the minimum value of the cost and the corresponding number of units to be
produced.
Solution :
C = 56 - 8x + x2
Differentiate with respect to x,
The minimum value of cost =
56–32+16
= 40
The corresponding number of units
produced = 4
Example 6.28
The total cost function of a firm
is C(x) = x3/3 – 5x2
+ 28x + 10 where x is the output. A
tax at the rate of ₹ 2 per
unit of output is imposed and the producer adds it to his cost. If the market
demand function is given by p = 2530
– 5x, where p is the price per unit of output, find the profit maximizing the
output and price.
Solution :
Total revenue: R =
p x
= (2530 – 5x)x
= 2530x–5x2
Tax at the rate ₹ 2 per x unit = 2x.
= 2530 – 5(50)
= ₹ 2280.
Example 6.29
The manufacturing cost of an item
consists of ₹ 1,600 as
over head material cost ₹ 30 per
item and the labour cost ₹ a (x2 /100) for x
items produced. Find how many items be produced to have the minimum average
cost.
Solution :
As per given information for
producing x units of certain item C(x) = labour cost + material cost +
overhead cost
AC is minimum at x = 400
Hence 400 items should be
produced for minimum average cost.
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