Question

Verify Rolle's theorem for the following function on the indicated interval f(x) = sin x + cos x on [0, π/2] ?

Answer 1

 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 

\[\left[ 0, \frac{\pi}{2} \right]\] and differentiable on \[\left( 0, \frac{\pi}{2} \right)\] .

Also,

\[f\left( \frac{\pi}{2} \right) = f\left( 0 \right) = 1\]

Thus,\[f\left( x \right)\] satisfies all the conditions of Rolle's theorem.

Now, we have to show that there exists

\[c \in \left( 0, \frac{\pi}{2} \right)\] such that  \[f'\left( c \right) = 0\] .

We have

\[f\left( x \right) = \sin x + \cos x\]

\[ \Rightarrow f'\left( x \right) = \cos x - \sin x\]

\[\therefore f'\left( x \right) = 0\]
\[ \Rightarrow \cos x - \sin x = 0\]
\[ \Rightarrow \tan x = 1\]
\[ \Rightarrow x = \frac{\pi}{4}\]

Thus,\[c = \frac{\pi}{4} \in \left( 0, \frac{\pi}{2} \right)\] such that

\[f'\left( c \right) = 0\] .

​Hence, Rolle's theorem is verified.