Question

The value of $x$ for which the tangents to the curves $y=x\cos x,y=\frac{\sin x}{x}$ are parallel to the axis of $x$ are roots of

• A. $\sin x=x,\tan x=x$
• B. $\cot x=x,\sec x=x$
• C. $\cot x=x,\tan x=x$
• D. $\tan x=x,\cot x=x$

Answer 1

The correct option is C $\cot x=x,\tan x=x$

Let $y=f\left(x\right)=x\cos x\Rightarrow f^{\prime }\left(x\right)=\cos x-x\sin x$
and $y=g\left(x\right)=\frac{\sin x}{x}\Rightarrow g^{\prime }\left(x\right)=\frac{x\cos x-\sin x}{x^2}$
Since tangents are parallel to $x-$ axis
so, $f^{\prime }\left(x\right)=0\Rightarrow x=\cot x$
and $g^{\prime }\left(x\right)=0\Rightarrow x=\tan x$