f(x) is decreasing -> f'(x) is
negative
If f(x) is a continuous function, then we can tell whether it is increasing or decreasing by examining the sign of its derivative f'(x). Specifically, if f(x) is decreasing over some interval, then its derivative f'(x) must be negative over that interval.
This can be seen mathematically because the derivative of a function f(x) gives us the slope of the tangent line to the function at each point x. If f(x) is decreasing, then the slope of the tangent line is negative, indicating a downward slope. Therefore, f'(x) must be negative.
Conversely, if f'(x) is negative over some interval, we can conclude that f(x) is decreasing over that interval.
In short, if f(x) is decreasing, then f'(x) is negative.
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