f(x) is decreasing -> f'(x) is
negative
To determine the relationship between the function f(x) and its derivative f'(x), we need to recall the definition of a decreasing function: a function is decreasing if its values decrease as its input increases.
Now, let’s consider the derivative f'(x), which is defined as the slope of the tangent line to the graph of f(x) at any given point x. If f(x) is decreasing, we know that the graph of f(x) slopes downward as we move from left to right. This means that the derivative f'(x) is negative at each point on the graph of f(x): as x increases, f'(x) becomes more negative.
In summary, if f(x) is decreasing, f'(x) is negative at each point on the graph of f(x). We can express this relationship mathematically as:
f(x) is decreasing -> f'(x) < 0 for all x in the domain of f(x)
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