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प्रश्न
Verify Rolle's theorem for the following function on the indicated interval f (x) = \[{e^{1 - x}}^2\] on [−1, 1] ?
उत्तर
The given function is \[f\left( x \right) = e^{1 - x^2}\] .
Since exponential function is everywhere continuous and differentiable, \[e^{1 - x^2}\] is continuous on \[\left[ - 1, 1 \right]\] and differentiable on \[\left( - 1, 1 \right)\] .
Also,
Thus, \[f\left( x \right)\] satisfies all the conditions of Rolle's theorem.
\[ \Rightarrow f'\left( x \right) = - 2x e^{1 - x^2}\]
\[ \Rightarrow - 2x e^{1 - x^2} = 0\]
\[ \Rightarrow x = 0\]
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