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Q)

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

• 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 $c / f' (c) = 0 \: and \: c \in (-1, 1)$
$-2x\: e^{1-x^2}=0 \Rightarrow x = 0 \: \in ( -1, v ) c = 0$