Question

The interval in which the function $f(x)=\frac{\log (7+x)}{\log (3+x)}(x>0)$ decreases is

• A. $\left(0,\frac{7}{3}\right)$
• B. $\left(0,\frac{3}{7}\right)$
• C. $(0,1)$
• D. $(0,\infty )$

Answer 1

Answer: (D)

If the given function $f(x)=\frac{\log (7+x)}{\log (3+x)},x>0$ is
decreasing function then $f^{\prime }(x)<0$ $\Rightarrow \frac{\log (3+x)\frac{1}{(7+x)}-\frac{1}{(3+x)}\log (7+x)}{[\log (3+x){]}^2}<0$
$\Rightarrow \frac{\log (3+x)\frac{1}{(7+x)}-\frac{1}{(3+x)}\log (7+x)}{[\log (3+x){]}^2}<0$
$\Rightarrow (3+x{)}^{(3+x)}<(7+x{)}^{(7+x)}$ , it is true for every value of $x>0$
$\Rightarrow x\in (0,\infty )$