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Differentiation is an essential concept in calculus which involves finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. When we differentiate a function, we are calculating the slope of the tangent to the curve at any given point on the function's graph.
A key characteristic of a derivative is that it can tell us about rising or falling behavior of the function:

• If the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.
• If the derivative is zero, the function is neither increasing nor decreasing at that point and could potentially be constant.

When a function's derivative is consistently zero over an interval, like in this problem, it implies that the function's slope is flat, and thus, the function doesn't change value. This leads us to conclude that the function must indeed be constant on that interval.

A constant function is one where the output value is the same for any input value on its domain. In simple terms, if you input any number into a constant function, the output will always be the same number.
In mathematical terms, for a constant function, say, a function \(f(x)\), there exists a constant \(c\) such that \(f(x) = c\) for every \(x\) in its domain. This means there are no changes, increases, or decreases along its graph. Instead, the graph is just a straight horizontal line.This concept of a constant function directly ties to the condition \(f'(x) = 0\) for all \(x\) over the interval given in the problem. Essentially, a derivative yielding zero implies no change, hence a constant function.

Interval notation is a way of writing subsets of the real number line. It is a concise method to express a range of values without having to make an exhaustive list.

Types of Intervals:

• Closed Interval: Notated \([a, b]\), it includes all the numbers between \(a\) and \(b\), as well as the endpoints themselves.
• Open Interval: Notated \((a, b)\), it includes all the numbers between \(a\) and \(b\) but not the endpoints.
• Half-Open Intervals: They can be either \([a, b)\) or \((a, b]\), including one endpoint but not the other.

In this problem, the function is continuous on a closed interval \([a, b]\) and differentiable on an open interval \((a, b)\). This distinction ensures that the function behaves nicely across its domain and any required derivatives behave as expected on the open interval. Understanding these intervals is crucial, especially since certain mathematical theorems, like the Mean Value Theorem, have conditions based on such interval types.