When f ‘(x) is negative, f(x) is
When the derivative of a function, denoted as f ‘(x), is negative, it indicates that the function f(x) is decreasing in value as x increases in the given interval
When the derivative of a function, denoted as f ‘(x), is negative, it indicates that the function f(x) is decreasing in value as x increases in the given interval. In other words, the function’s output, or y-values, are getting smaller as the input, or x-values, are getting larger. This suggests that the graph of the function has a negative slope in that interval.
To better understand this concept, let’s consider an example. Suppose we have a function f(x) = 3x^2 – 4x. The derivative of this function is f ‘(x) = 6x – 4.
Now, if we evaluate the derivative at a specific point, say f ‘(2), and find that f ‘(2) = 8, we can conclude that the function is increasing at x = 2. However, if we evaluate the derivative at another point, f ‘(-1), and find that f ‘(-1) = -10, we can determine that the function is decreasing at x = -1.
In summary, when the derivative f ‘(x) is negative, the function f(x) is decreasing in value as x increases. This information helps us understand the behavior and trends of the function graphically and analytically.
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