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Question
f (x) = x2/3 on [−1, 1] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
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Solution
The given function is \[f\left( x \right) = x^\frac{2}{3}\] on \[\left[ - 1, 1 \right]\] .
The domain of f is given to be \[\left[ - 1, 1 \right]\] .
Differentiating \[f\left( x \right)\] with respect to x, we get
\[f'\left( x \right) = \frac{2}{3} x^{- \frac{1}{3}}\]
We observe that at \[x = 0\] \[f'\left( x \right)\] is not defined.
Hence, Rolle's theorem is not applicable for the given function.
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