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Question
Function f(x) = ax is increasing on R, if
Options
a > 0
a < 0
0 < a < 1
a > 1
Solution
a > 1
\[f\left( x \right) = a^x \]
\[f'\left( x \right) = a^x \log a\]
\[\text { Given: f(x) is increasing on R .} \]
\[ \Rightarrow f'\left( x \right) > 0\]
\[ \Rightarrow a^x \log a > 0\]
\[ \Rightarrow a^x > 0 \left( \text { Logarithmic function is defined for positive values of a } \right)\]
\[\text { We know,} \]
\[ a^x \log a > 0\]
\[\text { It can be possible when} a^x > 0 \text { and } \log a > 0 or a^x < 0 and \log a < 0 \left( \text { Not possible, logarithmic function is defined for positive values of a } \right)\]
\[ \Rightarrow \log a > 0\]
\[ \Rightarrow a > 1\]
\[\text { So,f (x) is increasing when }a> 1 .\]
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