Page 335 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
1
The critical points are x and x = 1
3
dy
These points divide x-axis into intervals on which is either positive or negaitve.
dx

1 1
3

1 1 dy 1 1 3
When x , say , 3 4 1 0
3 4 dx 16 4 16
1 2 dy 4 2 1
When x say , 3 4 1 0
3 3 dx 4 3 3
4 dy 16 4
When x > 1 say , 3 4 1 1 0
3 dx 9 3
dy 1
Since changes from positive to neative at x , the function has a local maxima at
dx 3
1
x .
3
Similarly function has a minima at x = 1

(b) W e can w rite y = x – x 2/3
5/3
dy 5 2 2 1 1 1 5x 2
= x 3 x 3 = x 3 5x 2
dx 3 3 3 3x 1 3

dy 2
0 at x and uindefined at x = 0. These are two critical points.
dx 5
0 2
5

dy 7
When x < 0, say x = –1, 0
dx 3
2 1 dy 1
When 0 x say x , 0
5 5 dx 1 1 3
3
5
2 3 dy 1
When x say x , 0
5 5 dx 3 1 3
3
5
2
Thus there is local maxima (of the type given in Figure 13.2(b)) at x = 0 and minima at x .
5

x
x
1. If the function f(x) is continuous at the point x = a, and lim f ( ) and lim f ( ) are
Notes x a – x a
both infinite with opposite signs, then the graph of f(x) has a cusp at x = a. Note
that the graph of the function given in example 1(b) above, has a cusp at x = 0.
x
2. If lim f ( ) and lim f ( ) are both infinite with same signs, then the graph of f(x)
x
x a – x a
1
has a vertical tangent at x = a. Note that ( )f x x has a vertical tangent at x = 0.
3
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