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Question 1 :

Prove that thefunction f(x) = loge xis increasing on (0, ∞).

Answer 1 :

Let x1,x2 ∈ (0, ∞)

We have, x1 2

⇒ loge x1 < loge x2

⇒ f(x1) < f (x2)

So, f(x) is increasing in (0, ∞)

Question 2 : Prove that the function f(x) = loga x is increasing on(0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1.

Answer 2 :


Question 3 :

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

Answer 3 :

Given,

f (x) = ax + b, a > 0

Let x1,x2 ∈ R and x1 > x2

⇒ ax1 > ax2 for some a > 0

⇒ ax1 + b> ax2 + b for some b

⇒ f(x1) > f(x2)

Hence, x1 >x2 ⇒ f(x1) > f(x2)

So, f(x) is increasing function of R

Question 4 :

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

Answer 4 :

Given,

f (x) = ax + b, a < 0

Let x1,x2 ∈ R and x1 > x2

⇒ ax1 < ax2 for some a > 0

⇒ ax1 + b < ax2 + b for some b

⇒ f(x1) < f(x2)

Hence, x1 >x2⇒ f(x1) < f(x2)

So, f(x) is decreasing function of R

Question 5 :

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Question 8 :

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Question 9 :

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 View All Questions and Answers of RD Chapter 17- Increasing and Decreasing Functions Ex-17.1 
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