When f ‘(x) changes from negative to positive, f(x) has a
relative minimum
local minimum at x.
When the first derivative, f'(x), changes sign from negative to positive at a point x, it means that the function f(x) is increasing in a small interval around x. This also tells us that f(x) is decreasing on the left side of x, and increasing on the right side of x. Therefore, the function has a local minimum at x, since the function decreases slightly before x and increases slightly after x.
To visualize this, imagine a graph of f(x) where the function is initially decreasing before x, then reaches a low point at x, and then starts increasing again. This is a local minimum, and it occurs precisely when f'(x) changes from negative to positive.
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