Local Minimum: How The Function’S Slope Determines Its Lowest Point

When f ‘(x) changes from negative to positive, f(x) has a

relative minimum

When the derivative of a function, f ‘(x), changes from negative to positive at a specific point x, it means that the slope of f(x) is changing from decreasing to increasing. This indicates that the function is changing from decreasing to increasing, and thus it has a local minimum at that point.

A local minimum is a point on a function where the function reaches its lowest value in a specific region. It’s considered local because there may be other points on the function that are lower, but only in a smaller, isolated region.

To visualize this, imagine a graph of a hill. As you climb up the hill, the slope becomes steeper and steeper (f ‘(x) is decreasing). But once you reach the top of the hill, the terrain levels out and the slope becomes less steep (f ‘(x) is increasing). This flat region at the top of the hill is the local minimum of the function.

Overall, when f ‘(x) changes from negative to positive, this indicates a transition from decreasing to increasing slope, and thus the function has a local minimum at that point.

More Answers: