If there is a critical point between intervals where a function is increasing on both intervals, is there a relative max/min at that point?
To determine if there is a relative maximum or minimum at a critical point between intervals where a function is increasing on both intervals, we need to apply the First Derivative Test.
The First Derivative Test states that if a function is differentiable on an interval except at isolated points, and if the derivative changes from positive to negative at a critical point, then the function has a relative maximum at that point. Conversely, if the derivative changes from negative to positive at a critical point, then the function has a relative minimum at that point.
To apply the First Derivative Test, follow these steps:
1. Identify the critical point: Find the point where the derivative of the function equals zero or is undefined. This point is called the critical point.
2. Determine the values of the derivative on either side of the critical point: Evaluate the derivative of the function at a point slightly smaller than the critical point and at a point slightly larger than the critical point.
3. Analyze the sign changes in the derivative: If the derivative changes from positive to negative as you move from left to right across the critical point, then there is likely a relative maximum at that point. If the derivative changes from negative to positive, then there is likely a relative minimum at that point.
4. Verify the conditions: Confirm that the function is differentiable on the intervals and at the critical point, and ensure that the critical point is not an endpoint of the interval.
Therefore, if there is a critical point between intervals where a function is increasing on both intervals, and the derivative changes sign at that critical point, there is likely a relative maximum or minimum at that point.