#### 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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Solution 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 ? Concept: Maximum and Minimum Values of a Function in a Closed Interval.