A local maximum is a point on a function where the output value (y-coordinate) is greater than or equal to all the output values of neighboring points, within a certain interval. In other words, it is a high point on the function that is surrounded by lower points on both sides. Similarly, a local minimum is a low point on a function that is surrounded by higher points on both sides.
To find intervals that contain local maximums or minimums, we need to look for changes in the direction of the function. Specifically, we need to look for places where the function changes from increasing to decreasing (a local maximum) or decreasing to increasing (a local minimum).
A graph of a function can be used to identify local maximums by looking for the highest point in a given interval. This can be done by finding the highest point on the graph within the interval, or by finding the x-value where the derivative of the function equals zero (since a local maximum occurs where the derivative changes from positive to negative).
Here Are Some Steps You Can Follow To Identify Local Maximums From A Graph:
Look for high points on the graph: Start by identifying the highest points on the graph of the function. These points may be clearly marked on the graph, or you may need to estimate them based on the shape of the curve.
Check neighboring points: Once you have identified a potential local maximum, check the output values of the neighboring points on the graph. If the output values of the neighboring points are lower than the potential local maximum, then it is likely that the point is a local maximum.
Determine the interval: To determine the interval that contains the local maximum, look at the x-values of the neighboring points. The interval will be the range of x-values between the two neighboring points.
Verify the result: Once you have identified the interval that contains the local maximum, you can verify your result by calculating the derivative of the function within that interval. If the derivative changes sign from positive to negative, then the point is a local maximum.
In conclusion, identifying intervals that contain local maximums requires analyzing the direction of the function and looking for changes in its behavior. This can be done by examining the graph of the function and looking for high points that are surrounded by lower points. To verify the result, you can calculate the derivative of the function within the interval and check for a change in sign.
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