Page 354 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

Notes
10 10
or ln = 1 or e
p p
or p = 10.e –1

10
Also x = 400ln p 400ln e 400

–1
Thus x = 400 and p = 10.e at maximum revenue. To find price elasticity of demand, we
write.
ln x = ln 400(ln10 ln )p

d ln x 1 400 1 400
dp = 400 ln10 lnp p 10 p
400ln
p

d log p 1
Also =
dp p
d ln x 1 400 1
= p
d lnp 10 p 10
400ln ln
p p

1 10
= 1 at p
lne e

Example: If p = f(x) is an inverse demand function, find the level of output at which total
revenue is maximum. Show that total revenue will always be a maximum if demand curve is
downward sloping and concave from below. Is it possible to have maxima of total revenue if the
demand curve is convex from below? Discuss.
Solution:
d TR
TR = xf(x) f x xf x 0 for maxima
dx
f x p
f x =
x x
Since p and x are always positive, this implies that total revenue is maximum only if f x < 0

2
d TR = 2f x xf x
dx 2 0 (second order condition)
2f x
f x < , which is always satisfied if f x 0 .
x

2f x
When the demand curve is convex from below such that 0 f x , it is possible to have
x
maxima of total revenue.

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