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Q.
The minimum value of the function $f\left(x\right)=\frac{tan x}{3 + 2 tan x}, \, \forall x\in \left[0 , \frac{\pi }{2}\right)$ is
NTA AbhyasNTA Abhyas 2022
Solution:
If $x \in\left[0, \frac{\pi}{2}\right), \tan x \in[0, \infty)$
Let, $\tan x=t$
$
\begin{array}{l}
\text { i.e. } y=\frac{t}{3+2 t}, t \in[0, \infty) \\
\Rightarrow 3 y+2 t y=t \\
\Rightarrow t=\frac{3 y}{1-2 y} \geq 0 \\
\Rightarrow y \in\left[0, \frac{1}{2}\right)
\end{array}
$
Hence, the minimum value of the function $=0$