f'(x) changes from negative to positive
f(x) has a relative minimum
When f'(x) changes from negative to positive, it means that the derivative of the function is changing from negative to positive. This indicates that the function is increasing in that region.
In mathematical terms, if a function f(x) is differentiable, then its derivative can be either positive, negative or zero at any given point. If the derivative is positive at a point, then the function is increasing at that point. And if the derivative is negative at a point, then the function is decreasing at that point.
For example, let’s consider the function f(x) = x^2. The derivative of this function is f'(x) = 2x. Here, if f'(x) changes from negative to positive at some point c, then it means that the slope of the tangent line at point c is increasing. As a result, the function is increasing in that region.
Similarly, if the derivative changes from positive to negative at some point, then it means that the function is decreasing in that region. And if the derivative is zero at some point, then it indicates a critical point, which could be a local maxima or a local minima of the function.
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