When f ‘(x) is negative, f(x) is
When f ‘(x) is negative, it means that the derivative of the function f(x) with respect to x is negative
When f ‘(x) is negative, it means that the derivative of the function f(x) with respect to x is negative. In other words, the function is decreasing in that interval.
This implies that as x increases, the corresponding values of f(x) decrease. Graphically, this would be represented by a downward slope of the function.
For example, if we have the function f(x) = x^2, the derivative f ‘(x) = 2x. When x is negative, f ‘(x) is negative, indicating that the function is decreasing. As x increases, the values of f(x) decrease, as seen by the graph of the parabola opening downward.
In general, when f ‘(x) is negative, it means that for any two points, say a and b, in that interval where f ‘(x) is negative, if a is to the left of b (a < b), then f(a) will be greater than f(b) (f(a) > f(b)). This indicates a decreasing trend in the function.
It is important to note that this information holds for continuous functions. Discontinuous functions may have points where the derivative is negative but the function itself may not exhibit a clear decreasing trend.
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Understanding Derivatives: The Formal Definition and Calculation Steps for Finding the Derivative of a FunctionUnderstanding the Alternate Definition of Derivative: A Precise Explanation of Rate of Change in Mathematics
Understanding Positive Derivatives: Implications and Graphical Interpretation of f ‘(x) > 0