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Question
What are the values of 'a' for which f(x) = ax is increasing on R ?
Solution
\[f\left( x \right) = a^x \]
\[f'\left( x \right) = a^x \log a\]
\[\text { Given }: f(x) \text { is increasing on R } . \]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow a^x \log a > 0\]
\[\text { Logarithmic function is defined for positive values of a } . \]
\[ \Rightarrow a > 0\]
\[ \Rightarrow a^x > 0\]
\[\text { We know,} \]
\[ a^x \log a > 0\]
\[\text{ It can be possible when } a^x > 0 \text { and } \log a > 0 \text { or }a^x < 0 \text { and } \log a < 0 . \]
\[ \Rightarrow \log a > 0\]
\[ \Rightarrow a > 1\]
\[\text { So, }f(x)\text { is increasing when }a> 1 .\]
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