Page 337 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
3
Example: Find relative maxima and minima of the function y = x – 4x – 3x + 2.
2
Also find absolute maxima/minima in [0, 4].
Solution:
2
3
Given the function y = x – 4x – 3x + 2, we have
dy
= 3x – 8x – 3 = 0, for maxima or minima.
2
dx
2
Rewriting this equation as 3x – 9x + x – 3 = 0
or 3x(x – 3) + (x – 3) = 0 or (x – 3)(3x + 1) = 0
1
x = 3 or x
3

2
d y
Further, = 6x – 8 = 10 > 0, when x = 3
dx 2
1
and = – 10 < 0, when x
3
1
Thus, the function has a minima at x = 3 and maxima at x .
3
To find maxima/minima in [0, 4], we note that there is only one stationary point x = 3 in the
given interval.

2
Let f(x) = x – 4x – 3x + 2
2
f(0) = 2
f(3) = 27 – 36 – 9 + 2 = –16
f(4) = 64 – 64 – 12 + 12 = –10

Function has absolute maxima at x = 0, and aboslute minima at x = 3

1
Example: Show that the function y x has one maximum and one minimum value
x
and later is larger than the former. Draw a graph to illustrate this.
Solution:
1
Given y = x , we have
x

dy 1 x 2 1 2
= 1 0 or 0 x 1
dx x 2 x 2
or x = ±1 are the stationary points.

2
d y 2
Further, = 3 , which will be positive when
dx 2 x

330 LOVELY PROFESSIONAL UNIVERSITY