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RD Chapter 17 Increasing and Decreasing Functions Ex 17.1 Solutions

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

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