If f (. c) ≤ f (x) for every x in the domain of f, then the point (c, f (. c)) is a local minimum.
To determine if the point (c, f(c)) is a local minimum, we need to consider the behavior of the function f(x) in its neighborhood
To determine if the point (c, f(c)) is a local minimum, we need to consider the behavior of the function f(x) in its neighborhood.
If f(c) ≤ f(x) for every x in the domain of f, this means that the value of f(c) is less than or equal to the values of f(x) for all x in the domain of f. In other words, f(c) is either the smallest value or one of the smallest values of the function within its domain.
Now, in order to be a local minimum, the point (c, f(c)) should have a lower value than all the other points in its immediate neighborhood.
Since we have f(c) ≤ f(x) for every x in the domain of f, it implies that f(c) is less than or equal to all the other values of f(x) in its domain. Therefore, within its immediate neighborhood, there will not be any point (x, f(x)) with a lower value than (c, f(c)).
This satisfies the condition for a point to be a local minimum, as it is the smallest value within its immediate neighborhood. Therefore, we can conclude that if f(c) ≤ f(x) for every x in the domain of f, then the point (c, f(c)) is indeed a local minimum.
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