Question

Verify Rolle's Theorem for the function $f\left(x\right)=e^x(\sin x-\cos x)$ on $[\frac{\pi }{4},\frac{5\pi }{4}]$

Answer 1

$f\left(x\right)=e^x(\sin x-\cos x),xϵ[\frac{\pi }{4},\frac{5\pi }{4}]$
Sine, cosine and exponential function are always continuous.
$\therefore$ Given function is continuous in $[\frac{\pi }{4},\frac{5\pi }{4}]$
Differentiating w.r. to $x$, we get
$f^{\prime }\left(x\right)=e^x(\cos x+\sin x)+(\sin x-\cos x)e^x$
$=e^x[\cos x+\sin x+\sin x-\cos x]$
$=2e^x\sin x$
Which exists for all $x$.
$f\left(\pi /4\right)=e^{\pi /4}(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}})=0$
and $f\left(5\pi /4\right)=e^{5\pi /4}(-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}})=0$
$\therefore f(\pi /4=f(5\pi /4)=0$
$\therefore$ The given function statisfies all three condition of Rolle's theorem.
For maxima or minima
$f^{\prime }\left(x\right)=0$
$2e^x\sin x=0$
$\sin x=0$
$x=n\pi +(-1{)}^n(0)$
$x=n\pi$
$x=\pi$
$\because \pi$ lies between $[\frac{\pi }{4},\frac{5\pi }{4}]$ so Rolle's theorem is verified.