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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] ? - 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) = 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?

#### APPEARS IN

Solution for question: 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. For the courses CBSE (Commerce), CBSE (Arts), PUC Karnataka Science, CBSE (Science)
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