PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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Solution for Verify Rolle'S Theorem for the Following Function on the Indicated Interval F(X) = Sin X + Cos X on [0, π/2] ? - PUC Karnataka Science Class 12 - Mathematics

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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 
\[\left[ 0, \frac{\pi}{2} \right]\] 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.
  Is there an error in this question or solution?
Solution Verify Rolle'S Theorem for the Following Function on the Indicated Interval F(X) = Sin X + Cos X on [0, π/2] ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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