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