Question

Verify Rolle's theorem for the function $f\left(x\right)=\sin x+\cos x-1$ in the interval $[0,\frac{\pi }{2}]$

Answer 1

Since sine and cosine bring continuous lies in interval $[0,\frac{\pi }{2}]$
$f\left(x\right)=\sin x+\cos x-1$
$f\left(0\right)=0+1-1$
$f\left(0\right)=0$
$f\left(\frac{\pi }{2}\right)=1+0-1$
$f\left(\frac{\pi }{2}\right)=0$
$f\left(0\right)=f\left(\frac{\pi }{2}\right)$
$\therefore$ Rolle's theorem are satisfied
$f^{\prime }\left(x\right)=\cos x-\sin x$
So, there must exist some $c\in (0,\frac{\pi }{2})$ such
that $f^{\prime }\left(c\right)=0$
$f^{\prime }\left(c\right)=\cos c-\sin c$
$\cos c-\sin c=0$
$\cos c=\sin c$
$\therefore$ $c=\frac{\pi }{4}$
Thus $c=\frac{\pi }{4}\in (0,\frac{\pi }{2})$
$\therefore$ Rolle's theorem is verified