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Q)

Verify Rolle’s theorem for the function $ f(x)=\Large e^{1-x^2}$in the interval [-1,1]

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A)
Toolbox:
  • Check whether the function is continuous or not in the given closed interval [ a, b ]
  • Check whether it is differentiable or not in the given open interval ( a, b )
  • Check whether $ f(a)=f(b)$
  • Then find c in $ ( a, b) / f' (c) = 0$
$ e^{1-x^2} $ is continuous in [ -1, 1 ]
$ f'(x) = -2x\: e^{1-x^2}$
$ \Rightarrow f(n) $ is differentiable in ( -1, 1 )
$ \Rightarrow \exists $ atleast one point $ c / f' (c) = 0 \: and \: c \in (-1, 1)$
$ -2x\: e^{1-x^2}=0 \Rightarrow x = 0 \: \in ( -1, v ) c = 0$

 

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