Page 342 - DMTH201_Basic Mathematics-1

Unit 13: Maxima and Minima

Notes
Figure 13.7

2
Example: Show that the polynomial y = ax + bx + cx + d has only one point of inflexion.
3
Under what conditions
(a) The curvature changes from: (i) convex to concave and (ii) concave to convex?
(b) The point of inflexion is stationary?
Solution:
dy
2
3
(a) y = ax + bx + cx + d = 3ax + 2bx + c
2
dx
2
d y b
Further, = 6ax + 2b = 0 for point of inflexion x
dx 2 3a
2
d y
Since = 0 at a single value, there is only one point of inflexion.
dx 2
(i) For change of curvature from convex to concave, we must have
3
d y
= 6a < 0 a < 0
dx 3
(ii) Similarly, if a > 0, the curvature will change from concave to convex.
(b) The point of inflexion is said to be stationary if

dy b b 2 2b 2
= 3ax + 2bx + c = 0 at x 3a c 0
2
dx 3a 9a 2 3a
b 2 2b 2 b 2
or c = 0 or c 0 or b = 3ac
2
3a 3a 3a
1 x –x
Example: If y (e e ) show that
2
(a) y(x) = y(–x)
(b) y has a minima at x = 0
(c) The function has no point of inflexion.
Solution:

1 –x x 1 x –x
(a) y (– ) x (e e ) (e e ) y ( ) x
2 2

LOVELY PROFESSIONAL UNIVERSITY 335