The solution set of the inequation $3^x+3^{1-x}-4 \lt 0$ contained in $\mathbf{R}$ is

• A

  $(1,2)$
• B

  $(1,3)$
• C

  $(0,2)$
• Correct answer
  D

  $(0,1)$

Solution

$\begin{aligned} & 3^x+3^{1-x}-4 \lt 0 \\ \Rightarrow & 3^x+\frac{3}{3^x}-4 \lt 0 ; \operatorname{Le} t 3^x=t \Rightarrow t+\frac{3}{t}-4 \lt 0\end{aligned}$ $\Rightarrow t^2-4 t+3 \lt 0 \quad\left[\because t=3^x \neq 0\right]$ $\Rightarrow(t-1)(t-3) \lt 0$
$\begin{aligned} & \Rightarrow 1 \lt t \lt 3 \Rightarrow 1 \lt 3^x \lt 3 \Rightarrow 0 \lt x \lt 1 \\ & \Rightarrow x \in(0,1)\end{aligned}$

Hence, the correct answer is option: 'D'.