Question

The function $f(x) = \frac{ln (\pi + x)}{ln (e + x)}$ is

• A. increasing on $(0, \infty )$
• B. decreasing on $(0, \infty )$
• C. increasing on $(0, \pi /e)$, decreasing on $( \pi /e, \infty )$
• D. decreasing on $0, \pi /e)$, increasing on $(\pi /e, \infty )$

Answer 1

Answer: (B)

We have $f\left(x\right) = \frac{\text{In} \left(\pi+x\right)}{\text{In}\left(e+x\right)}$
$ \therefore f'\left(x\right) = \frac{\left(\frac{1}{\pi+x}\right) \text{In}\left(e+x\right) - \frac{1}{\left(e+x\right)} \text{In}\left(\pi+x\right)}{\left[\text{In} \left(e +x\right)\right]^{2}} $
$= \frac{\left(e+x\right)\text{In} \left(e+x\right)-\left(\pi+x\right) \text{In}\left(\pi+x\right)}{\left(e+x\right)\left(\pi+x\right)\left(\text{In}\left(e+x\right)\right)^{2}} $
< 0 on $(0, \infty)$ since $1 < e < \pi $
$\therefore \, f(x)$ decreases on $(0 , \infty).$