f’<0
f is decreasing
If f’ is less than zero, it means that the slope of the function f is negative at that point. In other words, the function is decreasing at that point.
To better understand this, imagine a graph of f(x). If the slope of the graph is negative (i.e. f’ <0), then the function is falling as x increases. This means that as x increases, the value of the function is decreasing. For example, let's say we have the function f(x) = x^2 - 3x. The derivative of this function is f'(x) = 2x - 3. If we plug in a value of x such as x = 2, we get f'(2) = 2(2) - 3 = 1. Since f'(2) is positive (greater than zero), we know that the graph of f is increasing at x = 2. Conversely, if we plug in a value of x such as x = 1, we get f'(1) = 2(1) - 3 = -1. Since f'(1) is negative (< 0), we know that the graph of f is decreasing at x = 1. This means that as x moves from left to right on the graph, the value of the function f(x) decreases at x = 1.
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