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Solution for Verify Rolle'S Theorem for the Following Function on the Indicated Interval F(X) = 4sin X on [0, π] ? - CBSE (Commerce) Class 12 - Mathematics

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Question

Verify Rolle's theorem for the following function on the indicated interval f(x) = 4sin x on [0, π] ?

Solution

The given function is \[f\left( x \right) = 4^{ sin \ x}\].

Since sine function and exponential function are everywhere continuous and differentiable,  \[f\left( x \right)\] is continuous on \[\left[ 0, \pi \right]\] and differentiable on \[\left( 0, \pi \right)\] .

Also,

\[f\left( \pi \right) = f\left( 0 \right) = 1\]
Thus, 
\[f\left( x \right)\] satisfies all the conditions of Rolle's theorem.
Now, we have to show that there exists
\[c \in \left( 0, \pi \right)\] such that \[f'\left( c \right) = 0\] .
We have 

\[f\left( x \right) = 4^{sin \ x } \]

\[ \Rightarrow f'\left( x \right) = 4^{sin x} \left( \cos x \right)\log4\]

\[\therefore f'\left( x \right) = 0\]

\[ \Rightarrow 4^{sin x} \left( \cos x \right)\log4 = 0\]

\[ \Rightarrow 4^{ sin x } \cos x = 0\]

\[ \Rightarrow \cos x = 0\]

\[ \Rightarrow x = \frac{\pi}{2}\]

Thus, 

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

Hence, Rolle's theorem is verified.

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Solution Verify Rolle'S Theorem for the Following Function on the Indicated Interval F(X) = 4sin X on [0, π] ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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