## Book: RD Sharma - Mathematics (Volume 1)

### Chapter: 17. Increasing and Decreasing Functions

#### Subject: Maths - Class 12th

##### Q. No. 6 of Exercise 17.2

Listen NCERT Audio Books to boost your productivity and retention power by 2X.

6
##### Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0.

Given:- Function f(x) = loga x , 0 < a < 1

Theorem:- Let f be a differentiable real function defined on an open interval (a,b).

(i) If f’(x) > 0 for all , then f(x) is increasing on (a, b)

(ii) If f’(x) < 0 for all , then f(x) is decreasing on (a, b)

Algorithm:-

(i) Obtain the function and put it equal to f(x)

(ii) Find f’(x)

(iii) Put f’(x) > 0 and solve this inequation.

For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.

Here we have,

f(x) = loga x, 0 < a < 1

As given 0 < a < 1

log(a) < 0

and for x > 0

Therefore f’(x) is

f’(x) < 0

Hence, condition for f(x) to be decreasing

Thus f(x) is decreasing for all x > 0

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39