By the 1st derivative test, a point is a relative maximum when
f’ changes from positive to negative
The first derivative test is used to determine the relative extrema (maximum and minimum points) of a function. According to the first derivative test, a point is a relative maximum when the derivative changes sign from positive to negative at that point.
More specifically, suppose f(x) is a function that is differentiable on an interval containing a critical point c. We can apply the first derivative test to determine whether c is a relative maximum or minimum by looking at the sign of f'(x) on either side of c.
If f'(x) changes from positive to negative at c, then c is a relative maximum. This is because the function is increasing to the left of c and decreasing to the right of c, which means that c represents a peak in the function.
On the other hand, if f'(x) changes from negative to positive at c, then c is a relative minimum. This is because the function is decreasing to the left of c and increasing to the right of c, which means that c represents a valley in the function.
If f'(x) does not change sign at c, then the test is inconclusive and we must use a higher-order derivative test to determine if c is a relative maximum, minimum, or saddle point.
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