Question

Which of the following is/are true for $f\left(x\right)=\frac{4x+3}{x^2+1}$

• A. Total number of real root of equation $f\left(x\right)=0$ is $1$
• B. Range of $f\left(x\right)$ is $[-1,4]$
• C. Equation $f\left(x\right)=2$ will have two real and distinct roots
• D. Total number of real roots of equation $f\left(x\right)=0$ are $3$

Answer 1

Answer: (A), (B), (C)

Equation $f\left(x\right)=2$ will have two real and distinct roots

$f\left(x\right)=\frac{4x+3}{x^2+1}$
For $x=0,y=3$
For $y=0,x=-\frac{3}{4}$
For $x\to \pm \infty ,y\to 0$
$\Rightarrow f^{\prime }\left(x\right)=\frac{(x^2+1)4-(4x+3)\left(2x\right)}{(x^2+1{)}^2}=\frac{-4x^2-6x+4}{(x^2+1{)}^2}$
$\Rightarrow f^{\prime }\left(x\right)=0$
$\Rightarrow x=-2,\frac{1}{2}$ are critical points of $f\left(x\right).$
Sign of $f^{\prime }\left(x\right):$
Graph of $f\left(x\right):$