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Theorems for Differentiability

Verify Roll's...

Question

Verify Roll's theorem for the following functions
$f (x) = \frac{sin x}{e^{x}}, x \in [0, π]$ .

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Solution

We have
$\begin{matrix} f (x) = \frac{sin x}{e^{x}}, x \in [0, π] f (0) = 0 a n d f (π) = 0 \Rightarrow t h e r e w i l l b e a t l e a t o n e c i n (0, π) f o r w h i c h f^{'} (c) = 0 N o w, f^{'} (x) = e^{- x} cos x - sin x e^{- x} f^{'} (x) = e^{- x} (cos x - sin x) f^{'} (\frac{π}{4}) = 0 0 < \frac{π}{4} < π H e n c e, R o l l^{'} s t h e o r e m i s v e r i f i e d . \end{matrix}$

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