Understanding the Relationship between the Derivative and Increasing/Decreasing Functions

When f ‘(x) is increasing, f(x) is

When the derivative, f ‘(x), of a function f(x) is increasing, it means that the rate of change of f(x) with respect to x is increasing as we move along the x-axis

When the derivative, f ‘(x), of a function f(x) is increasing, it means that the rate of change of f(x) with respect to x is increasing as we move along the x-axis. In other words, the slope of the tangent line to the graph of f(x) is increasing.

To understand what this implies about the function f(x), we need to consider different cases:

1. If f ‘(x) is strictly increasing:
– When f ‘(x) is positive, it means that f(x) is increasing. The function is getting larger as x increases.
– When f ‘(x) is negative, it means that f(x) is decreasing. The function is getting smaller as x increases.

2. If f ‘(x) is constant:
– If f ‘(x) is positive, then f(x) is increasing at a constant rate. The function is getting larger at the same rate as x increases.
– If f ‘(x) is negative, then f(x) is decreasing at a constant rate. The function is getting smaller at the same rate as x increases.

In summary, when f ‘(x) is increasing, it indicates that the function f(x) is either increasing or decreasing, depending on the sign of f ‘(x). If f ‘(x) is positive, f(x) is increasing, and if f ‘(x) is negative, f(x) is decreasing. If f ‘(x) is constant, then f(x) is either increasing or decreasing at a constant rate, again depending on the sign of f ‘(x).

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