1 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Chapter 6: Differentiation, Points of Inflexion Curvature and Applications u Curvature: definitions: concave up and concave down: Slide 2, 3 u Worked Example 6.27 (b): Figure Slide 4 u Points of inflexion: definitions and diagrams: Slide 5, 6 u Figure 6.34: Points of inflexion: Slide 7 u Applications: Slide 8, 9: Short-run production function Figure 6.36 u Applications: Slide 10, 11: Figure 6.37: Total cost and marginal cost

2 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Curvature u Concave up The curvature in the interval about a minimum: is described as concave up u Concave down The curvature in the interval about a maximum: is described as concave down

3 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Curvature u Concave up is sometimes described as convex towards the origin u Concave down is sometimes described as concave towards the origin

4 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Curvature positive for L > 0 u Worked Example 6.27 (b): Figure 6.33 Curve is convex towards the origin  Q0:

5 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Points of inflexion The point of inflexion is the point at which curvature changes along the interval in which curvature is concave up along the interval in which curvature is concave down at the point at which curvature changes

6 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Points of inflexion The point of inflexion is the point at which curvature changes along the interval in which curvature is concave down along the interval in which curvature is concave up at the point at which curvature changes

7 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Points of Inflexion u Figure 6.34 Points of inflexion at A1 and A2

8 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Applications of Points of inflexion u Production functions and marginal products of labour u Production Functions: See Figure u MP L is increasing up to the PoI: MP L is decreasing after the PoI u MP L is a maximum at the PoI: u The value of L at which MP L is maximized is the value of L at which the point of inflextion (at L = 10) occurs on the production function u The PoI is described as ‘The Law of diminishing returns to the factor labour’ :

9 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Point of inflexion on the production function Figure 6.36 Short-run production function, and APL functions (a) Short-run production function (b) MP L,APL functions Point of inflextion Maximum MP L Maximum APL Q = f ( L ) Q L APL MP L L L

10 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd Applications of Points of inflexion u The point of inflexion on usual total cost functions: u (a) The marginal cost is decreasing before, then u (b) increasing after, the point of inflexion u See Figure u MC is minimized at the point of inflexion (at Q = 10) on the total cost curve

11 Copyright©2001 Teresa Bradley and John Wiley & Sons Ltd u Figure 6.37: Total cost and marginal cost Figure 6.37 C MC Q C Minimum MC TVC TC Points of inflection (a) (b)