Page 359 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes AVC = 100 10 5 5 2 75
and MC = 100 20 5 3 5 2 75, at x = 5

Thus, MC = min. AVC
2
3
d C d MC d C 10
(iv) Since = 20 6x 0 and 6 0, at x , the total cost function has a
dx 2 dx dx 3 3
type II point of inflexion.
d MC 10
Since = 0 at x , MC is also minimum at this value.
dx 3
10 10 2 200
Also, min. MC = 100 20 3 66.67
3 3 2 3

Example: The cost of fuel consumed per hour in running a train is proportional to the
square of its speed (in kms per hour), and it costs 3,200 per hour at a speed of 40 kms per hour.
What is the most economical speed, if the fixed charges are 12,800 per hour?
Solution:
Let F be the cost of fuel and x be the speed of the train per hour. We are given that
F x 2 or F kx 2 , where k is a constant of proportionality.
3200
When x = 40, F is given to be 3,200, k 2.
1600

Thus we can write F 2x 2 , as the cost of fuel per hour of running the train when its speed is x
2
kms per hour. Now the total cost of running the train for x kms (per hour) is TC = 12,800 + 2x .
12800
x
Average cost AC = 2 .
x
The most economic speed will be that value of x which minimises AC.
d AC 12800
= 2 2 0, for minima or
dx x

or x 2 = 12800 6400 or x = 80 kms/hour.
2

Second order condition
2
d AC 25600 0 , when x = 80.
dx 2 = x 3
Thus, the second order condition for minima is satisfied.

Coefficients of a Cubic Total Cost Function

Let the cubic total cost function be TC = ax 3 bx 2 cx . d Therefore, the marginal cost function is
given by
d TC 2
MC = 3ax 2bx c
dx

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