Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
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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