Can you have 2 global minimums?

Global refers to the entire domain of the function. Global extrema are also called absolute extrema. There can be only one global maximum value and only one global minimum value.
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Do all functions have a global minimum and a global maximum?

While a function might not necessarily have global maximum or minimum, the Extreme Value Theorem tells us that a continuous function inside a closed interval must have a global maximum and a global minimum.
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Can infinity be a global maximum?

Note that the global maximum or minimum can also also on the boundary or points where the derivative dos not exist. absolute maximum as it goes to infinity for x → ∞. The function has a global minimum at x = 0 but the function is not differentiable there.
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Is it possible to have no absolute minimum or maximum for a function?

A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum. If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
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What is the absolute global minimum?

We say that f(x) has an absolute (or global) minimum at x=c if f(x)≥f(c) f ( x ) ≥ f ( c ) for every x in the domain we are working on. We say that f(x) has a relative (or local) minimum at x=c iff(x)≥f(c) f ( x ) ≥ f ( c ) for every x in some open interval around x=c .
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Can there be 2 global maximums?

Global refers to the entire domain of the function. Global extrema are also called absolute extrema. There can be only one global maximum value and only one global minimum value.
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Can a local minimum be a global minimum?

The maximum or minimum over the entire function is called an "Absolute" or "Global" maximum or minimum. There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.
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Does a continuous function always have an absolute maximum and minimum?

Theorem 1 If f is continuous on a closed interval [a, b], then f has both an absolute maximum value and an absolute minimum value on the interval. This theorem says that a continuous function that is defined on a closed interval must have both an absolute maximum value and an absolute minimum value.
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Can you have a local minimum at an endpoint?

The answer at the back has the point (1,1), which is the endpoint. According to the definition given in the textbook, I would think endpoints cannot be local minimum or maximum given that they cannot be in an open interval containing themselves.
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How do you find the global minimum and maximum?

Then to find the global maximum and minimum of the function:
  1. Make a list of all values of c, with a≤c≤b, a ≤ c ≤ b , for which. f′(c)=0, f ′ ( c ) = 0 , or. f′(c) does not exist, or. ...
  2. Evaluate f(c) for each c in that list. The largest (or smallest) of those values is the largest (or smallest) value of f(x) for a≤x≤b.
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Can you have an absolute max on an open interval?

For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. If the interval I is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over I.
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Is global maxima also local maxima?

The global maximum occurs at the middle green point (which is also a local maximum), while the global minimum occurs at the rightmost blue point (which is not a local minimum).
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What is the difference between local and global max and min?

A maximum or minimum is said to be local if it is the largest or smallest value of the function, respectively, within a given range. However, a maximum or minimum is said to be global if it is the largest or smallest value of the function, respectively, on the entire domain of a function.
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Does every continuous function have a global maximum?

We examine a fact about continuous functions. A function has a global maximum at , if for every in the domain of the function. A function has a global minimum at , if for every in the domain of the function. A global extremum is either a global maximum or a global minimum.
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Do endpoints count as critical points?

Critical points are usually defined as points where the first derivative vanishes, so no end points can be critical points (as there is no derivative).
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Can an endpoint be a relative max or min?

Relative extrema can certainly occur at endpoints of a domain. For instance, the function f(x) = x on the interval [0, 1] has a relative maximum at x = 1 and a relative minimum at x = 0.
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Can you take a derivative at an endpoint?

It says that the derivative takes on all values between the derivatives at the endpoints, and thus needs the one-sided derivatives at the endpoints to exist. Interestingly, Darboux's Theorem does not require the function to be continuous on the open interval between the endpoint.
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How do you find the global maximum and minimum on a closed interval?

The Closed Interval Method
  1. Find all critical numbers of f within the interval [a, b]. ...
  2. Plug in each critical number from step 1 into the function f(x).
  3. Plug in the endpoints, a and b, into the function f(x).
  4. The largest value is the absolute maximum, and the smallest value is the absolute minimum.
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Can a discontinuous function have a minimum?

Right: A discontinuous function y = f(x) on the closed interval [0, 3] that has both an absolute max and absolute min.
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Is a critical point always a maximum or minimum?

Determine whether each of these critical points is the location of a maximum, minimum, or point of inflection. For each value, test an x-value slightly smaller and slightly larger than that x-value. If both are smaller than f(x), then it is a maximum. If both are larger than f(x), then it is a minimum.
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Can a convex function have multiple local minima?

It's actually possible for a convex function to have multiple local minima, but the set of local minima must in that case form a convex set, and they must all have the same value. So, for instance, the convex function f(x)=max{‖x‖−1,0} has a minimum of 0 for all ‖x‖≤1.
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What is the difference between local minimizer and global minimizer?

A point x∗ ∈ Rn is called a local minimizer of the optimization problem minx∈Ω f(x), if there exists a neighbourhood N of x∗ such that x∗ is a global minimizer of the problem minx∈Ω∩N f(x). Date: January 2015. f(x∗) < f(x) whenever x ∈ Ω \ {x∗} satisfies x − x∗ ≤ ε.
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How do you prove a global minimum?

These global extrema can occur at critical points of f or at the boundary of the domain, where f is defined. Definition: A point a is called a global maximum of f if f(a) ≥ f(x) for all x. A point a is called a global minimum of f if f(a) ≤ f(x) for all x.
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What's a global maximum?

A global maximum, also known as an absolute maximum, the largest overall value of a set, function, etc., over its entire range. It is impossible to construct an algorithm that will find a global maximum for an arbitrary function.
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Can a function have both a local and absolute maximum at the same point?

A local maximum need not be an absolute maximum and an absolute maximum need not exist. f(-1) = -2 which is a local minimum and f(1)=2 is a local maximum. It's possible for there to be an x so that f(x)>2 and if so, that would be the absolute max on the interval.
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