Question:
If Rolle�s theorem for $f\left(x\right)= e^{x} \left(sinx - cosx\right)$ is verified on $[\pi/4$, $5 \pi/4]$, then the value of $c$ is
If Rolle�s theorem for $f\left(x\right)= e^{x} \left(sinx - cosx\right)$ is verified on $[\pi/4$, $5 \pi/4]$, then the value of $c$ is
Updated On: Jun 23, 2024
- $\pi/3$
- $\pi/2$
- $3\pi/4$
- $\pi$
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The Correct Option is D
Solution and Explanation
Given, $f (x) = e^{x} (sin \,x = cos\, x)$
on differentiating both sides w.r.t. , $x_{1}$ we get
$f'(x)=e^{x} \frac{d}{d x}(sin \,x-cos\, x)+(sin \,x-cos \,x) \frac{d}{dx}\left(e^{x}\right)$
[by using product rule of derivative]
$=e^{x}(cos\, x+sin \,x)+(sin\, x-cos \,x) e^{x}$
$=2 e^{x} \, sin \,x$
We know that, if Rolle's theorem is verified,
then their exist $c \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$, such that $f' (c)=0$
$\therefore 2 e^{c} \,sin\, c=0$
$\Rightarrow sin\, c=0$
$\Rightarrow c=\frac{\pi}{2} \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$
on differentiating both sides w.r.t. , $x_{1}$ we get
$f'(x)=e^{x} \frac{d}{d x}(sin \,x-cos\, x)+(sin \,x-cos \,x) \frac{d}{dx}\left(e^{x}\right)$
[by using product rule of derivative]
$=e^{x}(cos\, x+sin \,x)+(sin\, x-cos \,x) e^{x}$
$=2 e^{x} \, sin \,x$
We know that, if Rolle's theorem is verified,
then their exist $c \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$, such that $f' (c)=0$
$\therefore 2 e^{c} \,sin\, c=0$
$\Rightarrow sin\, c=0$
$\Rightarrow c=\frac{\pi}{2} \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.