Page 357 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes (ii) Find the value of L for which output is maximum.
(iii) Find the value of L at which the total product curve has a point of inflexion and verify that
MP is maximum at this point. What is the nature of the point of inflexion?
L
(iv) If the manufacturer sells the product at a uniform price of 10 per unit, find the maximum
total revenue product and associated level of L.
Solution:
x 2
(i) AP = 11 16L L
L L
dx 2
MP = 11 32L 3L
L dL
d AP L
We have = 16 2L 0 AP is maximum at L = 8.
dL L
d AP
2
Since L = 2 0 , the second order condition is satisfied.
dL 2
2
The maximum AP = 11 + 168 8 = 75
L
Further, MP when L = 8, is 11 + 32 8 3 8 = 75
2
L
Thus, AP = MP , when AP is maximum.
L L L
(ii) For maximum output:
dx
= 11 32L 3L 2 0
dL
or (11 – L)(1 + 3L) = 0, \ L = 11. The other value, being negative, is dropped.
2
d x
Since 2 32 6L 34 0 , the second order condition for maxima is satisfied.
dL
(iii) For point of inflexion:
2
3
d x 16 d x
dL 2 = 32 – 6L = 0 L = 3 5.33 and dL 3 = – 6 < 0.
Thus the point of inflexion is of type I, i.e. the curve changes from convex to concave from
below.
2
3
d x ( d MP L ) d x d 2 (MP L )
Since 2 = 0 and 3 = 2 0 ,
dL dL dL dL
MP is maximum at L = 5.33.
L
(iv) TRP = p.x = 10 11L 16L 2 L 3 .
L
Since TRP is a constant multiple of the production function, therefore, maxima of TRP will be
L L
at the same level of L where x is maximum. Thus, TRP will also be maximum at L = 11. The
L
3
maximum value = 10 11 11 16 11 2 11 = 7,260.
14.3 Minimisation of Cost
x
C F ( ) dC
x
If total cost C = F(x), then we can define AC , and MC = F ( ).
x x dx
Very often we are interested in finding the level of output that gives minimum AC. For minima
of AC, we have

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