If f'(x) has a change of sign and is always defined, then f(x) has either a
If f'(x) has a change of sign and is always defined, then f(x) has either a local maximum or a local minimum at the point where f'(x) changed sign
If f'(x) has a change of sign and is always defined, then f(x) has either a local maximum or a local minimum at the point where f'(x) changed sign.
To understand this, let’s first define a few terms:
1. Derivative (f'(x)): The derivative of a function f(x) measures its rate of change. It represents the slope of the tangent line to the graph of the function at any given point. When the derivative is positive, the function is increasing, and when it is negative, the function is decreasing.
2. Change of sign: When we say that the derivative has a change of sign, it means that the derivative changes from positive to negative or from negative to positive at a particular point on the graph of the function.
3. Local maximum/minimum: A local maximum occurs at a point on the graph of a function where the values of the function are greater than or equal to the values of the function at nearby points. Similarly, a local minimum occurs when the values of the function are smaller than or equal to the values of the function at nearby points.
Now, let’s consider the scenario when f'(x) has a change of sign and is always defined. This means that at a specific point on the graph, the slope of the tangent line changes from positive to negative or from negative to positive.
If f'(x) changes from positive to negative as we move along the x-axis, it suggests that the function is increasing up until that point and then starts decreasing afterward. This type of change indicates that f(x) has a local maximum at that specific point.
On the other hand, if f'(x) changes from negative to positive, it implies that the function is decreasing up until the point and then starts increasing afterward. This change suggests that f(x) has a local minimum at that specific point.
In summary, when f'(x) has a change of sign and is always defined, it tells us that the function f(x) has either a local maximum or a local minimum at the point where the derivative changed sign.
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