When f ‘(x) changes from negative to positive, f(x) has a
When a function’s derivative, f'(x), changes from negative to positive, it indicates that the function f(x) is increasing at that particular point
When a function’s derivative, f'(x), changes from negative to positive, it indicates that the function f(x) is increasing at that particular point. This means that as x increases, the corresponding values of f(x) are also increasing.
To understand this concept better, let’s consider an example. Suppose we have a function f(x) = x^2, and we want to determine when f ‘(x) changes from negative to positive.
First, let’s find the derivative of f(x) = x^2 using the power rule:
f'(x) = 2x
To find when f ‘(x) changes from negative to positive, we need to set f ‘(x) = 0 and solve for x:
2x = 0
x = 0
Now, let’s create a sign chart to analyze the behavior of f ‘(x) around x = 0.
|x | f ‘(x) |
|—-|——-|
|-∞ to 0| – |
|x = 0 | 0 |
|0 to ∞ | + |
From the sign chart, we can see that f ‘(x) is negative for x values less than 0 and positive for x values greater than 0. Therefore, when f ‘(x) changes from negative to positive, f(x) has an increasing slope at x = 0.
In the case of f(x) = x^2, this means that as x increases from negative values (e.g., -1, -2, etc.) to positive values (e.g., 1, 2, etc.), the corresponding values of f(x) will increase. For example, f(-2) = 4 and f(2) = 4, with f(x) values increasing as we move from left to right on the number line.
In summary, when f ‘(x) changes from negative to positive, it indicates that the function f(x) is increasing at that point.
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