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In calculus, the derivative of a function is a fundamental concept that measures how a function changes as its input changes. The derivative represents the rate of change or the slope of the function at any given point. For the function \(f(x) = a(1 - e^{-bx})\), the first derivative \(f'(x)\) is calculated using the chain rule, resulting in \(f'(x) = abe^{-bx}\).
This formula tells us the rate at which \(f(x)\) changes with respect to \(x\). Since exponential functions with negative exponents, like \(e^{-bx}\), always remain positive, it ensures that \(f'(x) > 0\) for all real \(x\) when \(a > 0\) and \(b > 0\).

• The positivity of \(f'(x)\) indicates an increasing function.
• Derivatives help in understanding various characteristics of functions, such as trends and rates of change.

Therefore, by analyzing \(f'(x)\), we can conclude that \(f(x)\) is always increasing.

Concavity describes the direction in which a function curves. A function is concave down if its graph opens downwards like a frown, and concave up if it opens upwards like a smile. To determine concavity, the second derivative \(f''(x)\) is examined. For our function, \(f''(x) = -ab^2 e^{-bx}\).
The sign of \(f''(x)\) provides insight into the concavity:

• If \(f''(x) < 0\), the function is concave down.
• Conversely, if \(f''(x) > 0\), the function is concave up.

In this case, since \(a > 0\) and \(b > 0\), the component \(-ab^2 e^{-bx}\) is inherently negative for all real \(x\).
Thus, it is established that the function \(f(x)\) does not change its concavity and is always concave down.

An increasing function is one where the value of the function grows as the input grows. It implies that as \(x\) moves to larger values, \(f(x)\) also moves to larger values. For a function to be classified as increasing over its entire domain, its first derivative must be non-negative for all values of \(x\).
For \(f(x) = a(1 - e^{-bx})\), the first derivative \(f'(x) = ab e^{-bx}\) assures us that it is positive, provided \(a > 0\) and \(b > 0\).

• A positive derivative means the function climbs upward as \(x\) increases.
• This growing trend is uniform for every point on the graph.

In summary, exploring the first derivative reveals \(f(x)\) to be increasing across the entire domain of \(x\).