APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES5 Marks
Question
What are the values of $'a'$ for which $f(x) = a^x $ is increasing on $R?$
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Answer
$\text{f}(\text{x})=\text{a}^\text{x}$
$\text{f}'(\text{x})=\text{a}^{\text{x}}\log\text{a}$
Given: $f(x)$ is increasing on $R$.
$\Rightarrow\ \text{f}'(\text{x})>0$
$\Rightarrow\text{a}^{\text{x}}\log\text{a}>0$
Logarithmic function is defined for positive values of $a.$
$\Rightarrow\text{a}>0$
$\Rightarrow\text{a}^\text{x}>0$
We know,
$\text{a}^{\text{x}}\log\text{a}>0$
It can be possible when $\text{a}^\text{x}>0$ and $\log\text{a}>0$ or
$\text{a}^\text{x}<0$ and $\log\text{a}<0.$
$\Rightarrow\log\text{a}>0$
$\Rightarrow\text{a}>1$
So, $f(x)$ is increasing when $a > 1.$
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