Verify Rolles thm for the function.
*F(x)= e^x sinx in [0,1]*
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Class 12
>>Maths
>>Continuity and Differentiability
>>Mean Value Theorems
>>Verify Rolle's theorem for the function
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Verify Rolle's theorem\quad for the function f(x)=e
−x
sinx,xϵ[0,π]
Hard
Solution
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f(x)=e
−x
sinx,x∈(0,π)
For Rolle's Theorem, f(0)=f(π) & f(x) must be continuous & differentiable over [0,π]
Let us check if f(0)=f(π)
⇒f(0)=e
−0
sin(0)=0
⇒f(π)=e
−π
sin(π)=0
Therefore, f(0)=f(π)
The function e
−x
& sinx are both continuous & differentiable over [0,π]
Therefore, Rolle's Theorem can be applied for the function given.
There exists
′
c
′
such that f
′
(c)=0
f
′
(x)=−e
−x
sinx+e
−x
cosx
f
′
(c)=−e
−c
sin(c)+e
−c
cos(c)=0⇒e
−c
[cosc−sinc]=0
⇒cosc=sinc⇒tanc=1⇒
c=
4
π
Hence, Rolle's Theorem is verified.