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A function has a "local minimum point" at a point p where there exists at least one positive number e having the property that the value v of the function for any point q for which the absolute value of q - p is greater than 0 but not greater than e, the value of the function at q is greater than or equal to the value at p.
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What is the lowest point on a graph or curve?
The lowest point on a graph or curve is known as the local minimum or global minimum, depending on its context. A local minimum is a point where the function value is lower than that of its immediate neighbors, while a global minimum is the absolute lowest point across the entire graph. This point often represents a minimum value of the function being graphed and can be identified using calculus techniques such as finding the derivative and setting it to zero.
What is the criteria for a function to have local minimum and maximum value?
A function has a local minimum or maximum at a point if the derivative at that point is zero (i.e., the first derivative test). Additionally, to determine whether it is a minimum or maximum, the second derivative test can be applied: if the second derivative is positive at that point, it indicates a local minimum, while a negative second derivative indicates a local maximum. If the second derivative is zero, further analysis may be required.
What is the definition of the Minimum of a function in Alegbra?
A global minimum is a point where the function has its lowest value - nowhere else does the function have a lower value. A local minimum is a point where the function has its lowest value for a certain surrounding - no nearby points have a lower value.
What is the math definition of minimum in math?
In mathematics, the minimum refers to the smallest value in a given set or function. For a set of numbers, the minimum is the least element among them. In the context of a function, the minimum point is where the function takes its lowest value within a specified domain. It can be classified as a global minimum (the lowest point over the entire domain) or a local minimum (the lowest point within a specific interval).
What is the lowest point on a graph in the domain of the function called?
The lowest point on a graph in the domain of the function is called the "minimum" or "global minimum" if it is the lowest point overall. If the lowest point is only the lowest within a certain interval, it may be referred to as a "local minimum." These points represent the values of the function where it attains its least value in the specified context.
Is a absolute minimum point also a local minimum point?
Yes. (But not the other way round - a local minimum is not necessarily an absolute minimum.)Yes. (But not the other way round - a local minimum is not necessarily an absolute minimum.)Yes. (But not the other way round - a local minimum is not necessarily an absolute minimum.)Yes. (But not the other way round - a local minimum is not necessarily an absolute minimum.)
What is the lowest point on a graph or curve?
The lowest point on a graph or curve is known as the local minimum or global minimum, depending on its context. A local minimum is a point where the function value is lower than that of its immediate neighbors, while a global minimum is the absolute lowest point across the entire graph. This point often represents a minimum value of the function being graphed and can be identified using calculus techniques such as finding the derivative and setting it to zero.
What is the criteria for a function to have local minimum and maximum value?
A function has a local minimum or maximum at a point if the derivative at that point is zero (i.e., the first derivative test). Additionally, to determine whether it is a minimum or maximum, the second derivative test can be applied: if the second derivative is positive at that point, it indicates a local minimum, while a negative second derivative indicates a local maximum. If the second derivative is zero, further analysis may be required.
What is the lowest point of the curve called?
The lowest point of a curve is called the "minimum." In mathematical terms, it represents the point where the function reaches its lowest value in a given interval. If the curve is part of a larger function, this minimum can be classified as a local minimum (lowest point in a small neighborhood) or a global minimum (lowest point across the entire function).
What are derivatives of minimus?
Derivatives of a minimum refer to the rates of change of a function at its minimum point. In calculus, at a local minimum, the first derivative is zero, indicating that the function is flat at that point. The second derivative is positive, confirming that the function is curving upwards, which characterizes a local minimum. Understanding these derivatives helps in optimization problems to identify and confirm minimum values of functions.
What is the difference between a local energy minimum and a global energy minimum in the context of energy optimization?
In energy optimization, a local energy minimum is a point where the energy is lower than in its immediate surroundings, but not necessarily the lowest overall. A global energy minimum is the point with the lowest energy value in the entire system. It is important to find the global minimum to achieve the most efficient energy optimization.
What is the definition of the Minimum of a function in Alegbra?
A global minimum is a point where the function has its lowest value - nowhere else does the function have a lower value. A local minimum is a point where the function has its lowest value for a certain surrounding - no nearby points have a lower value.
What is the math definition of minimum in math?
In mathematics, the minimum refers to the smallest value in a given set or function. For a set of numbers, the minimum is the least element among them. In the context of a function, the minimum point is where the function takes its lowest value within a specified domain. It can be classified as a global minimum (the lowest point over the entire domain) or a local minimum (the lowest point within a specific interval).
What is the lowest point on a graph in the domain of the function called?
The lowest point on a graph in the domain of the function is called the "minimum" or "global minimum" if it is the lowest point overall. If the lowest point is only the lowest within a certain interval, it may be referred to as a "local minimum." These points represent the values of the function where it attains its least value in the specified context.
What is the lowest point on a graph?
The lowest point on a graph is referred to as the minimum point or local minimum. It represents the smallest value of the function within a certain interval or over the entire domain if it's the absolute minimum. In graphical terms, it is the point where the graph changes direction from decreasing to increasing, often indicated by a vertex in quadratic functions or critical points in calculus.
Can points of inflection and extrema be at the same point?
Yes, points of inflection and extrema can occur at the same point on a function. A point of inflection is where the concavity of the function changes, while an extremum is a point where the function reaches a local maximum or minimum. In certain cases, such as the function (y = x^4) at (x = 0), the point can be both an inflection point and a local extremum, as the concavity changes while still being a minimum. However, this is not common and often requires specific conditions.
What is the extreme point called on a parabola?
The vertex, or maximum, or minimum.
