RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 17. Increasing and Decreasing Functions

Subject: Maths - Class 12th

Q. No. 2 of Exercise 17.1

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2

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decresing on (0, ∞), if 0 < a < 1.

case I

When a > 1


let x1,x2 (0,)


We have, x1<x2


⇒ loge x1 < loge x2


⇒ f(x1) < f(x2)


So, f(x) is increasing in (0,)


case II


When 0 < a < 1


f(x) = loga x


when a<1 log a < 0


let x1<x2


⇒ log x1 < log x2


⇒ []


⇒ f(x1) > f(x2)


So, f(x) is decreasing in (0,)


Chapter Exercises

Exercise 17.1
Exercise 17.2
MCQ

More Exercise Questions

1

Prove that the function f(x) = loge x is increasing on (0, ∞).
2

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decresing on (0, ∞), if 0 < a < 1.
3

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.
4

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.
5

Show that is a decreasing function on (0, ∞).
6

Show that decreases in the interval [0, ∞) and increases in the interval (-∞, 0].
7

Show that is neither increasing nor decreasing on R.
8

Without using the derivative, show that the function f(x) = | x | is

A. strictly increasing in (0, ∞)


B. strictly decreasing in (-∞, 0).
9

Without using the derivative show that the function f(x) = 7x - 3 is strictly increasing function on R.

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