Question

The number of points of inflection, number of point of intersections with $x-$axis and the number of points of local maximum for the curve $y=x+\tan x$ are given by $l,m$ and $n$ respectively $\forall x\in (-\frac{\pi }{2},\frac{\pi }{2}),$ then

• A. $l+m+n=1$
• B. $l=m=n$
• C. $l\ne m=n$
• D. $l+m+n=2$

Answer 1

The correct option is D $l+m+n=2$

$y=x+\tan x$ (odd function)
$x=0\Rightarrow y=0$
$\frac{dy}{dx}=1+{\sec }^{2}x>0$
$\Rightarrow$ strictly increasing function
$\frac{d^{2}y}{d{x}^2}=2{\sec }^{2}x\tan x=0$
$\Rightarrow x=0$ is point of inflection.
As $x\to {\frac{\pi }{2}}^{-}$ we have $y\to \infty$
and as $x\to -{\frac{\pi }{2}}^{+}$ we have $y\to -\infty$
$\therefore l=1,m=1,n=0$
$\Rightarrow l+m+n=1+1+0=2$