Question

Domain of the function $f(x)$ if $3^x+3^{f(x)}=$ minimum of $\varphi (t)$ where $\varphi (t)=$ minimum of $\left\{12t^3-15t^2+36t-25,2|\sin t|;2\le t\le 4\right\}$ is

• A. $(-\infty ,1)$
• B. $\left(-\infty ,{\log }_{3}e\right)$
• C. $\left(0,{\log }_{3}2\right)$
• D. $\left(-\infty ,{\log }_{3}2\right)$

Answer 1

Answer: (D)

Let $g(t)=2t^3-15t^2+36t-25$

$g^{\prime }(t)=6t^2-30t+36=6\left(t^2-5t+6\right)$
$=6(t-2)(t-3)=0$
$\Rightarrow t=2,3$
For $2\le t\le 4$
$g(t{)}_{\min }=g(3)$
$=2\times 27-15\times 9+36\times 3-25=2$
Also $2+|\sin t|\ge 2$
Hence minimum $\varphi (t)=2$
$\therefore 3^x+3^{f(x)}=2$
$\Rightarrow 3^{f(x)}=2-3^x$
$\Rightarrow 3^{f(x)}>0$
$\Rightarrow 2-3^x>0$
$\Rightarrow 3^x<2$
$\Rightarrow x<{\log }_{3}2$
$\therefore x\in \left(-\infty ,{\log }_{3}2\right)$