Understanding the Concept of Positive Derivatives: Analyzing the Increasing Behavior of Functions

When f ‘(x) is positive, f(x) is

When f ‘(x) is positive, it means that the derivative of the function f(x) with respect to x is positive

When f ‘(x) is positive, it means that the derivative of the function f(x) with respect to x is positive. This tells us that the function f(x) is increasing at that particular point.

To understand this, let’s consider the definition of the derivative: f ‘(x) = lim(h->0) [f(x+h) – f(x)] / h.

When f ‘(x) is positive, it means that as h approaches 0, the value of [f(x+h) – f(x)] / h is positive. This can happen in two ways:

1. Option 1: [f(x+h) – f(x)] is positive while h is positive.
In this case, for small positive values of h, the value of f(x+h) is larger than f(x), which indicates that the function f(x) is increasing.

2. Option 2: [f(x+h) – f(x)] is negative while h is negative.
In this case, for small negative values of h, the value of f(x+h) is smaller than f(x), which again indicates that the function f(x) is increasing.

In summary, when f ‘(x) is positive, it means that the function f(x) is increasing at that particular point.

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