Relative or Local minimum
at which it changes from decreasing to increasing
A relative (or local) minimum is a point on a function where the value of the function at the point is lower than the value of the function at nearby points. In other words, it is a point on a function where the function has its lowest value within a small interval around that point.
To determine whether a point on a function is a relative minimum, we can use the first or second derivative test.
First Derivative Test: We check the sign of the derivative of the function at the point. If the derivative changes sign from negative to positive at the point, then the point is a relative minimum. If the derivative changes sign from positive to negative at the point, then the point is a relative maximum.
Second Derivative Test: We take the second derivative of the function at the point. If the second derivative is positive, then the point is a relative minimum. If the second derivative is negative, then the point is a relative maximum.
It is important to note that a relative minimum may not be the minimum value of the entire function, as there may be lower values elsewhere. However, it is the lowest value of the function within a small interval around the point.
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