Relative minimum
Where first derivative = 0 & second derivative is positive
In mathematics, a relative minimum refers to a point on the graph of a function where the function has a lower value than the surrounding points in a specific interval or region. More formally, let’s consider a function f(x) defined over an interval I. If there exists a value c in I such that f(c) is smaller than or equal to f(x) for all x in a small neighborhood of c, then we say that f(c) is a relative minimum.
To better understand the concept of a relative minimum, it is essential to recognize the distinction between relative and absolute extrema. An absolute minimum refers to the lowest value of a function over its entire domain, while a relative minimum only concerns a specific interval. Relative extrema can occur at points where the function changes from decreasing to increasing (called a relative minimum) or from increasing to decreasing (called a relative maximum).
To identify the location of a relative minimum on a graph, we can use a variety of methods. One approach is to examine the behavior of the function’s derivative. If a function f(x) has a relative minimum at c, the derivative f'(x) will transition from positive to negative at x=c. This occurs because the function’s slope goes from increasing to decreasing around the point of the relative minimum.
Additionally, if c is a relative minimum, it means that f”(c) > 0. This condition indicates that the function has a concave-upward shape at the point c. In simple terms, the graph of the function curves like a U around the relative minimum.
To summarize, a relative minimum occurs at a point where the function takes on its lowest value within a specific interval. It is characterized by the derivative changing from positive to negative, and the second derivative being positive, resulting in a concave-upward shape.
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