When f ‘(x) changes from negative to positive, f(x) has a
When the derivative of a function, f ‘(x), changes from negative to positive, it indicates that the function, f(x), is increasing at that particular point, or in other words, the slope of the function is transitioning from negative to positive
When the derivative of a function, f ‘(x), changes from negative to positive, it indicates that the function, f(x), is increasing at that particular point, or in other words, the slope of the function is transitioning from negative to positive. This implies that the function is changing from decreasing to increasing or from a concave downward shape to a concave upward shape.
If we consider the graph of the function f(x), when f ‘(x) changes from negative to positive at a specific point, it means that the instantaneous rate of change (slope of the tangent line) is initially decreasing but then starts to increase. Visually, this is seen as a change in direction from a downward slope to an upward slope.
For example, let’s consider the function f(x) = x^2, which represents a simple quadratic curve. The derivative of this function, f ‘(x) = 2x, is positive for x values greater than 0 and negative for x values less than 0. At x = 0, the derivative changes sign from negative to positive. This indicates that f(x) = x^2 changes from decreasing to increasing at x = 0. So, the function has a minimum point at x = 0.
In summary, when f ‘(x) changes from negative to positive, it signifies that f(x) is changing from decreasing to increasing, implying a change in direction from a downward slope to an upward slope on the graph of the function.
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