Question

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

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

Answer 1

The correct option is B decreasing on $(0,\infty )$

We have $f\left(x\right)=\frac{ln(\pi +x)}{ln(e+x)}$
$\therefore f^{\prime }\left(x\right)=\frac{\left(\frac{1}{\pi +x}\right)ln(e+x)-\frac{1}{(e+x)}ln(\pi +x)}{\left[ln\right(e+x){]}^2}$
$=\frac{(e+x)ln(e+x)-(\pi +x)ln(\pi +x)}{(e+x)(\pi +x)\left(ln\right(e+x){)}^2}$
$<0on(0,\infty )since1<e<\pi$