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Is a local extremum also a global extremum?
A local extremum is not necessarily a global extremum. A local extremum is a point where the function reaches a maximum or minimum value within a small neighborhood, but it may not be the highest or lowest point in the entire function. A global extremum, on the other hand, is the highest or lowest point in the entire function. Therefore, a local extremum may or may not be a global extremum, depending on the behavior of the function in the larger context.
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Are all global extremum points also local extremum points of polynomial functions?
No, not all global extremum points are also local extremum points of polynomial functions. Global extremum points are the highest or lowest points on the entire function, while local extremum points are the highest or lowest points within a specific interval. A polynomial function can have global extremum points that are not local extremum points if the function continues to increase or decrease beyond the local extremum point.
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Is a local extremum also a global one?
No, a local extremum is not necessarily a global extremum. A local extremum is a point where a function reaches a maximum or minimum value within a small neighborhood, while a global extremum is the absolute maximum or minimum value of the function over its entire domain. A local extremum may or may not be the global extremum depending on the behavior of the function in the rest of the domain.
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What is the definition of a local extremum point?
A local extremum point is a point on a function where the function reaches a maximum or minimum value within a small neighborhood of that point. A local maximum occurs when the function reaches a peak at that point, and a local minimum occurs when the function reaches a valley at that point. These points are called "local" because they are only compared to nearby points, not the entire function.
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Are there third-degree functions that have exactly one local extremum?
Yes, there are third-degree functions that have exactly one local extremum. For example, the function f(x) = x^3 has exactly one local extremum at the point (0,0). This is because the function has a critical point at x=0 where the derivative changes sign from negative to positive, indicating a local minimum.
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What are extremum problems?
Extremum problems are mathematical problems that involve finding the maximum or minimum value of a function. These problems typically involve optimizing a certain quantity, such as maximizing profit or minimizing cost. Extremum problems are commonly encountered in various fields such as economics, engineering, and physics, where finding the optimal solution is crucial for decision-making and problem-solving. The process of solving extremum problems often involves using calculus techniques such as differentiation to find critical points where the function reaches its maximum or minimum value.
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What is a global extremum?
A global extremum is the highest or lowest value of a function over its entire domain. For a function with a global maximum, there is no other point in the domain where the function takes a higher value. Similarly, for a function with a global minimum, there is no other point in the domain where the function takes a lower value. Global extrema are important in optimization problems and can provide valuable information about the behavior of a function.
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How can one determine whether an extremum problem is a local minimum or maximum?
One can determine whether an extremum problem is a local minimum or maximum by using the second derivative test. If the second derivative at the critical point is positive, then the function has a local minimum at that point. If the second derivative is negative, then the function has a local maximum at that point. If the second derivative is zero, then the test is inconclusive and other methods, such as the first derivative test or analyzing the behavior of the function around the critical point, may be used to determine the nature of the extremum.
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