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, it means that the slope of the function is initially decreasing but then starts increasing at that point. In other words, the function is initially sloping downwards, but then it starts sloping upwards.
This change in sign of the derivative occurs at a critical point or a point of inflection. At this point, the second derivative of the function is zero or undefined, and the function changes its concavity.
If we graph the function f(x), the point where f'(x) changes from negative to positive will be a low point or a valley, also known as a local minimum. This point will be the lowest point of the curve in that particular area, and after passing this point, the function will start increasing.
This information can be useful in calculating the maximum or minimum value of a function, as well as determining any local extrema or inflection points. It can also help us to understand the behavior of the function and provide insight into its characteristics.
More Answers:
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