Increasing and Decreasing Functions — Maths STD 12 Science — Question

4. a > 1

Solution:

$\text{f}(\text{x})=\text{a}^\text{x}$

$\text{f}'(\text{x})=\text{a}^\text{x}\log\text{a}$

Given: f(x) is increasing on R.

$\Rightarrow\text{f}'(\text{x})>0$

$\Rightarrow\text{a}^\text{x}\log\text{a}>0$

$\Rightarrow\text{a}^\text{x}>0$

(Logarithmic function is defined for positive value of a)

We know,

$\Rightarrow\text{a}^\text{x}\log\text{a}>0$

It can be possible when $\text{a}^\text{x}>0$ and $\log\text{a}>0$ or $\text{a}^\text{x}<0$ and $\log\text{a}<0$

(Not possible, logarithmic function is defined for positive value of a)

$\Rightarrow\log\text{a}>0$

$\Rightarrow\text{a}>1$

So, f(x) is increasing when a > 1.