Relative Minimum
In mathematics, a relative minimum is a point on a function where the function has a lower value compared to points close to it, within a specific interval
In mathematics, a relative minimum is a point on a function where the function has a lower value compared to points close to it, within a specific interval.
More formally, let’s consider a function f defined on an interval [a, b]. If there exists a point c within this interval where f(c) is less than or equal to f(x) for all x within a small neighborhood of c, then we say that f has a relative minimum at c.
Geometrically, a relative minimum corresponds to a point on the graph of a function where the graph reaches a local low and then starts increasing again. It represents the lowest point on the function within a local region, but it may not be the absolute lowest point on the entire function.
To determine the presence of a relative minimum, we typically resort to the first derivative test or second derivative test.
– The first derivative test states that if c is a critical point of f (i.e., f'(c) = 0 or f'(c) does not exist), and the sign of f'(x) changes from negative to positive as we move from the left to the right of c, then f has a relative minimum at c.
– The second derivative test states that if c is a critical point of f and f”(c) > 0, then f has a relative minimum at c.
It’s important to note that the existence of a relative minimum does not guarantee the uniqueness of that minimum within the function. There can be multiple relative minima depending on the shape and behavior of the function.
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