Question

If ${}^{\prime }{c}^{\prime }$ is a value of $x$ for which Rolle's theorem holds for the function $f\left(x\right)=\sin \left(\frac{x}{2}\right)$ on the interval $[0,2\pi ]$, then which of the following is/are true

• A. Slope of the line $cx+2\pi y+7=0$ is $-\frac{1}{2}$
• B. Area of the circle, $x^2+y^2=c^2$ is ${\pi }^2$sq. units
• C. Period of $\tan \left(cx\right)=2$
• D. Period of $\sec \left(cx\right)=2$

Answer 1

Answer: (A), (D)

Period of $\sec \left(cx\right)=2$

Given,$f\left(x\right)=\sin \left(\frac{x}{2}\right)$ , $x\in [0,2\pi ]$
$f\left(0\right)=0,f\left(2\pi \right)=0$ and $f$ is continuous in $[0,2\pi ]$, differentiable in $(0,2\pi )$
$\therefore$ From,Rolle's Theorem $f^{\prime }\left(c\right)=0,$ for $c\in (0,2\pi )$
$\Rightarrow \frac{1}{2}\cos \left(\frac{c}{2}\right)=0$
$\Rightarrow c=\pi \in (0,2\pi )$
$a)$ Period of $\tan \left(cx\right)=\frac{\pi }{|c|}=\frac{\pi }{\pi }=1$
$b)$ Period of $\sec \left(cx\right)=\frac{2\pi }{|c|}=\frac{2\pi }{\pi }=2$
$c)$ Area of the circle, $x^2+y^2=c^2$ is $\pi c^2={\pi }^3$sq. units
$d)$ Slope of the line $cx+2\pi y+7=0$ is $-\frac{c}{2\pi }=-\frac{1}{2}$