Question

Function f(x) = a^x is increasing on R, if
(a) a > 0
(b) a < 0
(c) 0 < a < 1
(d) a > 1

Answer 1

(d) a > 1

$$\begin{gathered} f\left(x\right)=a^x \\ f\text{'}\left(x\right)=a^x\log a \\ \mathrm{Given}:f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{on}R. \\ \Rightarrow f\text{'}\left(x\right)>0 \\ \Rightarrow a^x\log a>0 \\ \Rightarrow a^x>0\left(\mathrm{Logarithmic}\mathrm{function}\mathrm{is}\mathrm{defined}\mathrm{for}\mathrm{positive}\mathrm{values}\mathrm{of}a\right) \\ \mathrm{We}\mathrm{know}, \\ {a}^x\log a>0 \\ \mathrm{It}\mathrm{can}\mathrm{be}\mathrm{possible}\mathrm{when}{a}^x>0\mathrm{and}\log a>0\mathrm{or}{a}^x<0\mathrm{and}\log a<0\left(\mathrm{Not}\mathrm{possible},\mathrm{logarithmic}\mathrm{function}\mathrm{is}\mathrm{defined}\mathrm{for}\mathrm{positive}\mathrm{values}\mathrm{of}\mathit{}a\right) \\ \Rightarrow \log a>0 \\ \Rightarrow a>1 \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing when}\mathit{\text{a}}\text{> 1}. \end{gathered}$$