In Exercises 59 and 60, give reasons for your answers. Let...

In Exercises 59 and $60,$ give reasons for your answers. Let $$ f(x)=(x-2)^{2 / 3} $$ $$ \begin{array}{l}{\text { a. Does } f^{\prime}(2) \text { exist? }} \\ {\text { b. Show that the only local extreme value of } f \text { occurs at } x=2 \text { . }} \\ {\text { c. Does the result in part (b) contradict the Extreme Value }} \\ {\text { Theorem? }} \\ {\text { d. Repeat parts (a) and (b) for } f(x)=(x-a)^{2 / 3}, \text { replacing } 2 \text { by } a \text { . }}\end{array} $$

In Exercises 59 and $60,$ give reasons for your answers.
Let
$$
f(x)=(x-2)^{2 / 3}
$$
$$
\begin{array}{l}{\text { a. Does } f^{\prime}(2) \text { exist? }} \\ {\text { b. Show that the only local extreme value of } f \text { occurs at } x=2 \text { . }} \\ {\text { c. Does the result in part (b) contradict the Extreme Value }} \\ {\text { Theorem? }} \\ {\text { d. Repeat parts (a) and (b) for } f(x)=(x-a)^{2 / 3}, \text { replacing } 2 \text { by } a \text { . }}\end{array}
$$

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must attain both a maximum and a minimum value somewhere in that interval. This theorem relies on the continuity and boundedness of the function, and does not necessarily depend on the differentiability of the function.

Fractional Exponent Functions

Functions defined by fractional exponents, such as those involving a term raised to the power of 2/3, often exhibit behaviors such as cusps or vertical tangents. These peculiarities can lead to situations where the derivative does not exist at certain points even though the function is continuous, affecting the analysis of local extrema and differentiability.

Local Extrema

Local extrema refer to points in the function’s domain where the function reaches a local maximum or minimum value. Determining local extrema typically involves identifying critical points by analyzing where the derivative is zero or undefined, and then using tests (such as the first or second derivative tests) to assess whether these points correspond to a local maximum or minimum.

Differentiability at a Point

This concept involves determining whether a function has a derivative at a specific point. A function is differentiable at a point if its left-hand and right-hand limits of the difference quotient exist and are equal. In cases where the function has a cusp or a vertical tangent at that point, these one-sided limits may differ, resulting in non-differentiability.

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