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Question
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?
Solution
The given function is \[f\left( x \right) = \sin x + \cos x\] .
Since \[\sin x \text { and } \cos x\] are everywhere continuous and differentiable, \[f\left( x \right) = \sin x + \cos x\] is continuous on
\[f\left( x \right) = \sin x + \cos x\]
\[ \Rightarrow f'\left( x \right) = \cos x - \sin x\]
\[\therefore f'\left( x \right) = 0\]
\[ \Rightarrow \cos x - \sin x = 0\]
\[ \Rightarrow \tan x = 1\]
\[ \Rightarrow x = \frac{\pi}{4}\]
Thus,\[c = \frac{\pi}{4} \in \left( 0, \frac{\pi}{2} \right)\] such that
\[f'\left( c \right) = 0\] .
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