relative minimum
When f ‘(x) changes from negative to positive, f(x) has a
In calculus, a relative minimum is a point on a graph where the function has a lower value than at all nearby points, but not necessarily the lowest value on the entire graph. A relative minimum point is also known as a local minimum.
Mathematically, a point c is considered to be a relative minimum of a function f if there exists some interval (a, b) containing c such that f(c) is less than or equal to f(x) for all x in the interval (a, b). This means that there may be other points on the graph with lower values, but they are outside the immediate vicinity of the relative minimum point.
To determine whether a point is a relative minimum, you can use the first derivative test or the second derivative test. The first derivative test involves examining the sign of the derivative before and after the point in question. If the derivative changes from positive to negative at the point, then the point is a relative minimum. The second derivative test involves examining the concavity of the function at the point. If the second derivative is positive, then the point is a relative minimum.
Relative minima are important in optimization problems, where the goal is to find the minimum or maximum value of a function. By identifying the relative minimum points, you can determine the region of the graph where the function is decreasing, and thus find the optimal solution within that range.
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