Question

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

$g'(t)=6 t^{2}-30 t+36=6\left(t^{2}-5 t+6\right)$
$=6(t-2)(t-3)=0$
$ \Rightarrow t=2,3$
For $2 \leq t \leq 4$
$g(t)_{\min }=g(3)$
$=2 \times 27-15 \times 9+36 \times 3-25=2$
Also $2+|\sin t| \geq 2$
Hence minimum $\phi(t)=2$
$\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)$