Chapter 6: Differentiation, Points of Inflexion
Curvature and Applications
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Curvature: definitions: concave up and concave down:
Slide 2, 3
Worked Example 6.27 (b): Figure 6.33. Slide 4
Points of inflexion: definitions and diagrams: Slide 5, 6
Figure 6.34: Points of inflexion: Slide 7
Applications: Slide 8, 9: Short-run production function
Figure 6.36
Applications: Slide 10, 11: Figure 6.37: Total cost and
marginal cost
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Curvature
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Concave up
The curvature in the interval
about a minimum: y   0
is described as concave up
y   0
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Concave down
The curvature in the interval
about a maximum: y   0
is described as concave down
y   0
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Curvature
Concave up is sometimes
described as convex towards the
origin
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y
y
y - f(x)
0
Concave down is sometimes
described as concave towards
the origin

y - f(x)
x
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
0
x
3
Curvature
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Worked Example 6.27 (b): Figure 6.33
Q
25
Q
L
Q = 25/L
d ( Q)
25
 2
dL
L
d 2 (Q) 50
 3
2
dL
L
0
L
Q   0: positive for L > 0
Curve is convex towards the origin
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Points of inflexion
The point of inflexion is the point at which curvature changes
y   0 along the interval in which curvature is concave up
y   0 along the interval in which curvature is concave down
y   0 at the point at which curvature changes
y   0
y   0
y   0
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Points of inflexion
The point of inflexion is the point at which curvature changes
y   0 along the interval in which curvature is concave down
y   0 along the interval in which curvature is concave up
y   0 at the point at which curvature changes
y   0
y   0
y   0
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Points of Inflexion
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Figure 6.34 Points of inflexion at A1 and A2
y   0
y   0
y   0
y   0
A1
Points of inflexion
A2
y   0
y   0
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Applications of Points of inflexion
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Production functions and marginal products of labour
Production Functions: See Figure 6.36.
MPL is increasing up to the PoI: MPL is decreasing after
the PoI
MPL is a maximum at the PoI:
The value of L at which MPL is maximized is the value of
L at which the point of inflextion (at L = 10) occurs on the
production function
The PoI is described as ‘The Law of diminishing returns to
the factor labour’ :
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Point of inflexion on the production function
Q
400
(a)
Short-run production function
Q = fL( )
350
300
250
Point of inflextion
200
150
100
50
L
22
20
18
16
14
12
10
8
6
4
2
0
0
Maximum MP L
MP L
30
APL
25
Maximum APL
20
AP L
15
10
5
-10
(b)
MP L , APL
22
20
18
16
14
12
10
-5
8
6
4
2
0
0
L
functions
-15
Figure 6.36 Short-run production function,
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
MP L
and APL functions
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Applications of Points of inflexion
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The point of inflexion on usual total cost functions:
(a) The marginal cost is decreasing before, then
(b) increasing after, the point of inflexion
See Figure 6.37.
MC is minimized at the point of inflexion (at Q = 10) on
the total cost curve
Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd
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Figure 6.37: Total cost and marginal cost
7000
6000
C
(a)
5000
TC
Points of inflection
4000
TVC
3000
2000
1000
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
(b)
C
700
MC
Minimum MC
600
500
400
300
200
100
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Q
Figure 6.37
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