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Question
What are the values of 'a' for which f(x) = ax is decreasing on R ?
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Solution
\[f\left( x \right) = a^x \]
\[f'\left( x \right) = a^x \log a\]
\[\text { Given }:f\left( x \right) \text { is decreasing on R }.\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in R\]
\[ \Rightarrow a^x \log a < 0, \forall x \in R\]
\[\text { Here, logaritmic function is not defined for negative values of a } . \]
\[ \Rightarrow a^x > 0 \]
\[ \therefore a^x \log a < 0 \text { can be possible when } \log a < 0, \forall x \in R . \]
\[ \Rightarrow 0 < a < 1\]
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