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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

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### 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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#### APPEARS IN

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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