Function f(x)=a^x is increasing on R, if: - Vidyadip

MCQ

Function $f(x)=a^x$ is increasing on $R$, if:

A
$a > 0$
B
$a < 0$
C
$0 < a < 1$
✓
$a > 1$

✓

Answer

Correct option: D.

$a > 1$

$\text{f}(\text{x})=\text{a}^\text{x}$
$\text{f}\ '(\text{x})=\text{a}^\text{x}\log\text{a}$
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 $a > 1.$

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