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Question
f(x) = 3 + (x − 2)2/3 on [1, 3] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Solution
The given function is \[f\left( x \right) = 3 + \left( x - 2 \right)^\frac{2}{3}\] Differentiating with respect to x, we get
\[f'\left( x \right) = \frac{2}{3} \left( x - 2 \right)^\frac{2}{3} - 1 \]
\[ \Rightarrow f'\left( x \right) = \frac{2}{3} \left( x - 2 \right)^\frac{- 1}{3} \]
\[ \Rightarrow f'\left( x \right) = \frac{2}{3 \left( x - 2 \right)^\frac{1}{3}}\]
Clearly, we observe that for x = 2
\[\in \left[ 1, 3 \right]\] \[f'\left( x \right)\] does not exist.
Therefore, \[f\left( x \right)\] is not derivable on \[\left[ 1, 3 \right]\]
Hence, Rolle's theorem is not applicable for the given function.
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