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Understanding the Chain Rule is crucial when dealing with composite functions. A composite function is a function made up of two or more functions, such as having a function nested within another. When you need to take the derivative of such a function, the Chain Rule comes to the rescue.

The Chain Rule states that if you have a composite function, say, \( g(f(x)) \), the derivative of this composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In other words, \( g'(f(x)) \times f'(x) \). This allows you to systematically take derivatives of complex expressions by breaking them down into simpler parts.

When working with trigonometric functions like cosine, it's important to know how to differentiate them. The derivative of the cosine function is quite straightforward. If you have \(f(x) = \text{cos}(x)\), its derivative is \(f'(x) = -\text{sin}(x)\).

It's a common mistake to forget the negative sign, but it is crucial for correct calculations. The negative sign indicates that the sine function, which represents the rate of change of the cosine function, decreases as the cosine function increases.

Critical points of a function are where its derivative is zero or undefined. These points are essential in studying the function's behavior, as they could indicate potential relative extrema - where the function reaches local maximums or minimums.

To find the critical points, you first derive the function and then solve for where the derivative equals zero. In the context of our exercise, setting \(f'(x) = -\text{sin}(\frac{\text{pi} x}{2}) \times \frac{\text{pi}}{2} = 0\) implies we need to find where the sine function equals zero. For the sine function, this occurs when its argument is an integer multiple of \(\text{pi}\). Hence, we determine that \(x = 2n\), for any integer \(n\), are the critical points of the given function.

Extrema refers to the maximums and minimums of a function. There are two types: absolute (or global) extrema, which are the highest and lowest points on the entire graph of the function, and relative (or local) extrema, which are the highest and lowest points within a specific range of the domain.

Identifying extrema is fundamental in optimization problems and in understanding the overall shape of a graph. To find relative extrema, one normally looks at critical points, as they indicate where the rate of change (slope) of the function is zero. In the exercise provided, after finding the critical points, we determine that the value of the derivative at these points is zero, suggesting potential relative extrema at these locations.