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The derivative of a function gives us information about the rate of change of the function's output relative to its input. When we find the derivative of a function, we are essentially catching a glimpse of how the function behaves in terms of increasing or decreasing at any point.

For the given function, \(f(t) = t^3 + t\), we use the power rule for differentiation, which states if \(f(t) = t^n\), then its derivative \(f'(t) = nt^{n-1}\). By applying this rule, the derivative becomes \(f'(t) = 3t^2 + 1\).

This derivative helps us assess the slope of the original function at any given point \(t\), but we also more importantly use it to find critical points by setting it to zero and solving for \(t\). In this exercise, as \(3t^2 + 1\) is always positive, it means \(f(t)\) never changes direction within the domain, indicating no critical points.

Absolute extrema are the highest and lowest values that a function takes on over a given interval. In simpler terms, they tell us the tallest peak and the steepest valley within that specific range.

To find these extreme values, it's common practice to first locate critical points by solving the derivative equal to zero, but in instances, like in this case, where no real solutions exist, we turn our attention to boundary points.

By evaluating the function at the boundary points \(t = -2\) and \(t = 2\), we calculate \(f(-2) = -10\) and \(f(2) = 10\) respectively. Since these are the only values we assess because there are no internal critical points, the absolute maximum is \(10\) at \(t = 2\), and the absolute minimum is \(-10\) at \(t = -2\).

This means the function reaches its highest and lowest values just at the ends of the interval.

Boundary points like \(-2\) and \(2\) outline the edges of the interval within which we're examining our function. When it comes to determining extrema, especially if the critical points fail to show up within the interval, examining these boundary points can provide crucial insights.

By plugging these points into the original function \(f(t) = t^3 + t\), we can gather the function's behavior at these specific points, allowing us to identify the absolute extrema. In the absence of critical points, these become our focal areas in determining the tall stature and depth of the function for the given span.

This method is especially handy since solving the derivative for zeros doesn't always yield real solutions, as evidenced by \(3t^2 + 1\). Therefore, scrutinizing boundary points ensures we don't miss the essential extrema influenced by the interval limitations.