Question

The solution set of $f^{\prime }\left(x\right)>g^{\prime }\left(x\right)$ where $f\left(x\right)=\left(\frac{1}{2}\right)5^{2x+1}$ and $g\left(x\right)=5^x+4x\log 5$ is

• A. $(1,\infty )$
• B. $(0,1)$
• C. $[2,\infty )$
• D. $(0,\infty )$

Answer 1

The correct option is D $(0,\infty )$

$f^{\prime }\left(x\right)=\left(\frac{1}{2}\right)5^{2x+1}\left(\log 5\right)\times 2=\left(\log 5\right){.5}^{2x+1}$

Also $g^{\prime }\left(x\right)=5^x\log 5+4\log 5$
So $\{x:f^{\prime }\left(x\right)>g^{\prime }\left(x\right)\}=\{x:\log {5.5}^{2x+1}>\left(\log 5\right)5^x+4\log 5\}$

$=\{x:5^{2x+1}>5^x+4\}=\{t=5^x:5t^2-t-4>0\}$
$=\{t=5^x:(5t+4)(t-1)>0\}$

$=\{t=5^x:t>1\text{or}t<-\frac{4}{5}\}=\{t=5^x:t>1\}$

$=(0,\infty )$