What are the values of 'a ' for which f(x) = a^x is decreasing on… - Vidyadip

Question

What are the values of $'a\ '$ for which $f(x) = a^x$ is decreasing 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 decreasing on $R.$
$\Rightarrow\ \text{f}'(\text{x}) < 0,\forall\ \text{x}\in\text{R}$
$\Rightarrow\text{a}^{\text{x}}\log\text{a} < 0,\forall\ \text{x}\in\text{R}$
Here, logarithmic function is not defined for negative values of $a.$
$\Rightarrow\text{a}^\text{x} > 0$
$\therefore\ \text{a}^{\text{x}}\log\text{a} < 0$
It can be possible when $\log\text{a} < 0,\forall\ \text{x}\in\text{R}.$
$\Rightarrow 0 < \text{a} > 1$

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