Let f(x)=(x-2)^2 / 3. a. Does f^'(2) exist? b. Show that the only local extreme value of f occur

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Let f(x)=(x-2)^2 / 3. a. Does f^'(2) exist? b. Show that the...

Let $f(x)=(x-2)^{2 / 3}$.
a. Does $f^{\prime}(2)$ exist?
b. Show that the only local extreme value of $f$ occurs at $x=2$.
c. Does the result in part (b) contradict the Extreme Value Theorem?
d. Repeat parts (a) and (b) for $f(x)=(x-a)^{2 / 3}$, replacing 2 by $a$.

Key Concepts

Extreme Value Theorem

This theorem states that a continuous function on a closed interval attains both its maximum and minimum values. It guarantees the existence of absolute extrema under these conditions, even if the function is not differentiable at the points where these extrema occur, clarifying that continuity and compactness are the key requirements rather than differentiability.

Differentiability at a Point

This concept addresses whether the derivative of a function exists at a specific point. A function may be continuous at a point yet fail to be differentiable there, particularly if the graph has a cusp or a corner. In such cases, even though the function behaves well in terms of overall value, the rate of change (slope) is not well-defined at the point.

Local Extremum

A local extremum (minimum or maximum) is a point where a function attains a value lower or higher than all nearby points. Identifying local extrema often involves testing where the derivative is zero or does not exist, and analyzing the behavior of the function around these critical points to determine if they are indeed minima or maxima.

Parameterized Function Transformation

Generalizing a function by introducing a parameter (for example, replacing a constant with a variable like 'a') allows the study of a family of functions with similar properties. This approach shows that many characteristics, such as differentiability and the existence of local extrema, can be analyzed in a broader context and are not confined to a single specific example.

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