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F(x)= e^-xsinx in (0,pi) verify rolles theorem

Dear Student,

Please refer to the following link.

https://www.meritnation.com/ask-answer/question/verify-rolles-theorem-for-f-x-exsinx-on-0-pi/continuity-and-differentiability/10395469

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  • F(0) = 1
  • F(pi) = 1
  • F(x) is continuous in [0,pi] and differentiable in (0,pi)
then According to Rolle's Theorem there exists "c" between 0 to pi such that F'(c) = 0.
F'(x) =e^(-xsinx) * (-sinx -xcosx) = 0
-sinx-xcosx = 0
c=x=0
indeed,F'(c) = 0 exists.
Hence Rolle's theorem is verified.
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