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To find the critical points of a function, we must first determine the derivative and set it equal to zero. This solution step is vital because the critical points help identify where the function might have maxima or minima within a given interval. For the function \( f(x) = x^{2/3}(20-x) \), we applied the product rule to find its derivative: \( f'(x) = \frac{2}{3}x^{-1/3}(20-x) - x^{2/3} \).

Critically, critical points occur where this derivative equals zero or where it is undefined. Solving \( f'(x) = 0 \) involves simplifying and reworking the derivative equation to identify the values of \( x \) where the function's rate of change momentarily becomes zero. These points suggest potential local peaks or troughs in the graph.

The concept of absolute maximum and minimum is central in calculus optimization. These terms refer to the highest and lowest values a function takes on a specified interval. After calculating critical points for our function \( f(x) = x^{2/3}(20-x) \), we must evaluate the function at these points and at the endpoints of the interval \([-1, 20]\).

By comparing these values, we determine the absolute highest and lowest points of the function. Remember that absolute extrema encompass endpoints because, in certain intervals, these can be where the function reaches its highest or lowest value due to either a natural trick of the graph's shape or the imposed limits.

The derivative of a function gives us a powerful tool to analyze the function's behavior, such as increasing or decreasing trends and locating extreme values. For the function \( f(x) = x^{2/3}(20-x) \), to find its derivative, we used the product rule. The product rule applies when differentiating a product of two functions and is expressed as \( \frac{d}{dx}(u \cdot v) = u'v + uv' \).

Given this function, \( x^{2/3} \) is \( u \) and \( 20-x \) is \( v \). Applying the rule correctly, we derived \( f'(x) = \frac{2}{3}x^{-1/3}(20-x) - x^{2/3} \).Solving \( f'(x) = 0 \) helps us find where the slope is zero — pinpointing possible maxima and minima.

Graphing utilities offer a visual representation of a function, helping detect features like peaks and valleys (which suggest possible maxima and minima), zeros, and overall shape. For our problem, graphing \( f(x) = x^{2/3}(20-x) \) over the interval \([-1, 20]\) aids in estimating the approximate location of these critical points before precise calculations.

By using a graphing utility, not only can we visually verify the presence of peaks and troughs, but we also gain an intuitive understanding of the function's behavior across the interval. After analytical calculus steps, graphing utilities also serve as a final check to ensure the accuracy of our answers by correlating points derived computationally with observed graphical features.