When f ‘(x) is negative, f(x) is
decreasing
When f'(x) is negative, it means that the function f(x) is decreasing. This means that the slope of the tangent line at any point on the curve is negative, which indicates that as x increases, the y-value of the function decreases. In other words, the function is dropping as we move from left to right on the x-axis.
For example, consider the function f(x) = -x^2. When we take the derivative of this function, we get f'(x) = -2x. This function is negative for all x except for 0 (where the slope is 0). This means that when x is less than 0, the function is decreasing. As x increases from negative infinity towards 0, the function is decreasing at a faster and faster rate. Similarly, as x increases from 0 to positive infinity, the function is also decreasing, but at a slower and slower rate.
In graphical terms, the function will be represented by a curve that is sloping downwards from left to right, and the tangent lines on this curve will all have negative slopes.
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