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Local extrema refer to points on a graph where a function takes either a local minimum or maximum value. These are points where the function changes direction. A local minimum is where the function value is less than values around it, while a local maximum is higher than its surrounding points.
To identify a local extremum, we often look at the derivative, as it tells us about the slope of the function. If the derivative changes sign, that indicates a potential extremum. However, sometimes the derivative doesn’t exist at certain points, yet those points can still be local extrema through other observations, such as graph analysis.
For the function \((x-2)^{2/3}\), there's a cusp at \(x=2\). The derivative is undefined at this point, but by examining the graph, we see that the function decreases as it approaches \(x=2\) from the left and increases as it moves away on the right. This indicates that \(x=2\) is a local minimum.

The Extreme Value Theorem (EVT) is a useful concept in calculus. It assures us that within a closed interval, a continuous function must achieve both a maximum and minimum value. This theorem requires that the interval is closed and bounded, and the function is continuous throughout.
For the function \(f(x) = (x-2)^{2/3}\), the theorem faces challenges since the natural domain is all real numbers, \(\mathbb{R}\), and lacks specified endpoints. Without such boundaries, EVT doesn't apply directly because there are no guaranteed maximum or minimum values over an unbounded domain.
Though \(x=2\) is a point of local minimum, it is identified not through EVT, but via observing the function's behavior near that point rather than across a bounded interval.

The power rule is a fundamental rule in calculus used to find the derivative of power functions, which are functions of the form \(x^n\) where \(n\) is any real number. The rule states: if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\).

• It simplifies finding slopes of tangent lines to graphs representing power functions, essential for understanding changes in behavior of the function at various points.
• Faster than other derivative methods, it's widely used in differentiating any term that is a pure power of \(x\).

Applying the power rule to \(f(x) = (x-2)^{2/3}\), we derive \(f'(x) = \frac{2}{3}(x-2)^{-1/3}\). Here, it shows how the function behaves differently around the point of interest, \(x=2\), where the derivative doesn't exist, highlighting the need for careful analysis when applying calculus rules to function behavior.