Question
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\frac{\text{x}}{2}-\sin\frac{\pi\text{x}}{6}\text{ on }[-1,0]$
$\text{f}(\text{x})=\frac{\text{x}}{2}-\sin\frac{\pi\text{x}}{6}\text{ on }[-1,0]$
We have
$\text{f}(\text{x})=\frac{\text{x}}{2}-\sin\frac{\pi\text{x}}{6}$ $\Rightarrow\text{f}'(\text{x})=\frac{1}{2}-\frac{\pi}{6}\cos\frac{\pi\text{x}}{6}$ $\therefore\ \text{f}'(\text{x})=0$ $\Rightarrow\frac{1}{2}-\frac{\text{x}}{6}\cos\frac{\pi\text{x}}{6}=0$ $\Rightarrow\cos\frac{\pi\text{x}}{6}=\frac{3}{\pi}$ $\Rightarrow\text{x}=\frac{-6}{\pi}\cos^{-1}\Big(\frac{3}{\pi}\Big)$ Thus, $\text{c}=\frac{-6}{\pi}\cos^{-1}\Big(\frac{3}{\pi}\Big)\in(-1,0)$ such that f'(c) = 0. Hence, Rolle's theorem is verified.Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.