Question

Function f(x) = a^x is increasing on R, if

पर्याय

• a > 0

• a < 0

• 0 < a < 1

• a > 1

Answer 1

a > 1

\[f\left( x \right) = a^x \]

\[f'\left( x \right) = a^x \log a\]

\[\text { Given: f(x) is increasing on R .} \]

\[ \Rightarrow f'\left( x \right) > 0\]

\[ \Rightarrow a^x \log a > 0\]

\[ \Rightarrow a^x > 0 \left( \text { Logarithmic function is defined for positive values of a } \right)\]

\[\text { We know,} \]

\[ a^x \log a > 0\]

\[\text { It can be possible when} a^x > 0 \text { and } \log a > 0 or a^x < 0 and \log a < 0 \left( \text { Not possible, logarithmic function is defined for positive values of a } \right)\]

\[ \Rightarrow \log a > 0\]

\[ \Rightarrow a > 1\]

\[\text { So,f (x) is increasing when }a> 1 .\]