Question

Consider a function $f ( x )=\frac{3 x + a }{ x ^2+3}$ which has greatest value equal to $\frac{3}{2}$.
The minimum value of $f ( x )$ is equal to

• A. $\tan \left(\frac{-\pi}{3}\right)$
• B. $\sin \left(\frac{-\pi}{6}\right)$
• C. $\cos \left(\frac{-\pi}{3}\right)$
• D. $\cot \left(\frac{\pi}{2}\right)$

Answer 1

Answer: (B)

$y=f(x)=\frac{3 x+3}{x^2+3} \Rightarrow x^2 y+3 y=3 x+3 \Rightarrow x^2 y-3 x+3 y-3=0$
As $x \in R$, so $D \geq 0$
$\Rightarrow 9-4 y(3 y-3) \geq 0 \Rightarrow 12 y^2-12 y-9 \leq 0 \Rightarrow 4 y^2-4 y-3 \leq 0 $
$\Rightarrow 4 y^2-6 y+2 y-3 \leq 0 \Rightarrow(2 y-3)(2 y+1) \leq 0$
Hence $y \in\left[\frac{-1}{2}, \frac{3}{2}\right]$
Hence minimum value of $f ( x )$ is $\frac{-1}{2}=\sin \left(\frac{-\pi}{6}\right)$