Relative maximum
Where first derivative = 0 & second derivative is negative
In mathematics, a relative maximum refers to the highest value that a function reaches within a specific interval or local region. It is also known as a local maximum.
To determine if a point on a graph is a relative maximum, we need to examine the values of the function in the surrounding area. If the function’s values decrease before and after the point in question, then it can be considered a relative maximum.
To provide a more formal definition, let’s say we have a function f(x). A point (a, f(a)) is considered a relative maximum if there exists an open interval (a – δ, a + δ) such that f(a) ≥ f(x) for all x in the interval, excluding a. In simpler terms, the value of the function at a relative maximum is greater than or equal to the values of the function at all nearby points.
It’s important to note that a relative maximum does not necessarily mean that the function has the highest value over its entire domain. It only refers to the highest value within a specific local region. A function can have multiple relative maxima and even a global maximum.
A helpful tool to find relative maxima is the first derivative test. According to this test, if a function has a relative maximum at a point, then the derivative of the function switches from positive to negative as we move from left to right of that point.
Overall, understanding relative maxima helps us analyze the behavior of functions and identify critical points where the function reaches its highest local values.
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