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Differentiability is a fundamental concept in calculus that signifies whether a function has a derivative at each point in its domain. A function is differentiable at a point if it is smooth or continuous there, without any abrupt changes in direction or sharp corners. When a function is differentiable across its entire domain, it means you can compute how fast it is changing at any point within that domain.
To determine if a function is differentiable, check its piecewise structure or look for points where the behavior alters.

• If a function is differentiable at a point, it means the derivative exists and is finite at that point.
• A differentiable function is necessarily continuous, but the reverse isn't always true; not all continuous functions are differentiable.

For example, the function given in our exercise, denoted as \( f \), is stated to be differentiable everywhere, enabling us to explore how the function behaves around specific points, such as where the derivative changes from negative to positive.

The concept of a derivative in calculus describes the rate at which a function is changing at any given point. It can be seen as a measure of the slope of the function at a particular point. The derivative, often denoted as \( f' \) or \( \frac{df}{dx} \), gives critical information about the function's instantaneous rate of change.

• If \( f'(x) > 0 \), the function \( f \) is increasing at \( x \).
• If \( f'(x) < 0 \), the function \( f \) is decreasing at \( x \).
• When \( f'(x) = 0 \), the function might have a local maximum, a local minimum, or a stationary point at \( x \).

In our exercise, the derivative of \( f \), \( f' \), is negative on the interval \((-\infty, 1)\) and positive on \((1, \infty)\). This information helps us determine the function's behavior, indicating it decreases towards \( x=1 \) and increases thereafter, forming a typical peak or valley scenario.

In calculus, identifying increasing and decreasing functions is crucial to understanding their overall behavior. An increasing function is one where the value of \( f(x) \) gets larger as \( x \) gets larger, and conversely, a decreasing function sees \( f(x) \) getting smaller as \( x \) increases. Derivatives help here by indicating this behavior based on their sign.
For example, let's consider the following:

• If \( f' < 0 \) on an interval, \( f \) is decreasing on that interval.
• If \( f' > 0 \) on an interval, \( f \) is increasing on that interval.

The function \( f \) in our problem is decreasing for all \( x \) less than 1 due to negatively-valued derivatives, and it's increasing for all \( x \) greater than 1 where the derivatives are positive. This behavior around \( x = 1 \) suggests the presence of a local minimum at that point, proving key for solving specific parts of the problem, such as establishing that \( f(x) \geq 1 \) everywhere by recognizing this minimum.