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Absolute value functions involve expressions within vertical bars, indicating that the value taken is always non-negative. They’re represented as |x|, where x could be any real number. This implies that the graph reflects the input x as itself when x is positive or zero, and as -x when x is negative, causing a characteristic 'V' shape in their graphs.

For the given function, when we deal with the absolute value of a function like \(f(x)=|x^2-1|\), it's crucial to consider both the inside function \(x^2-1\) flipping its sign depending on whether the inside expression is positive or negative. This is why the function is defined in pieces for \(x ≥ 1\) and \(x ≤ 1\). Understanding these pieces allows us to analyze the graph and differentiate effectively.

Derivatives represent the rate at which a function is changing at any given point and are foundational in calculus for identifying critical numbers, which highlight important features of the graph like slopes of tangents and rates of change. Derivatives are typically denoted as \(f'(x)\) for a function \(f(x)\).

In the given exercise, computing the derivative of the absolute value function creates two separate scenarios depending on the sign of the inside expression. For \(x ≥ 1\), the function's derivative is straightforward, \(f'(x) = 2x\), and for \(x ≤ 1\), the derivative is negative, \(f'(x) = -2x\), owing to the 'flip' caused by the absolute value. Knowing how to calculate and interpret derivatives is essential to understanding the behavior of a function.

Local maximum and minimum points are high and low spots in a function where the function value peaks or troughs relative to nearby points. These points are particularly interesting because they represent extremal values in a certain interval.

When analyzing \(f(x) = x^2 - 1\) for \(x ≥ 1\) and \(f(x) = 1 - x^2\) for \(x ≤ 1\), we use derivatives to find where the slope of the tangent line is horizontal—the function's rate of change being zero. This is generally done by setting the derivative equal to zero. In this context, the critical number \(x = 0\) is checked for being a local max or min by comparing the function values around it. As seen, \(x = 0\) has the same value for slight perturbations to the left and right, indicating it is neither, which can be a transitional point, known as an inflection point.

A point of inflection is a point on the curve of the graph of a function where the curve changes from being concave (curved upwards) to convex (curved downwards), or vice versa. In simpler terms, it signifies a change in the direction of curvature.

For the function \(f(x) = |x^2 - 1|\), the critical number found is \(x = 0\). Despite this point not being a local max or min, it's relevant because the concavity of the function changes at this point. The rate of change does not 'speed up' or 'slow down' in the usual sense of a maximum or minimum; instead, it 'switches direction'. This is a more sophisticated concept involving the second derivative, but even with first derivative analysis, students can observe this shift in the graph’s curvature. Understanding inflection points deepens one's comprehension of a function's geometry.