Explain why the first derivative can be used to determine over what intervals a function is increasing or decreasing?

The first derivative of a function \(f(x)\) represents its rate of change at any given point. Specifically, the sign of the derivative indicates the direction in which the function is changing. - If \(f'(x) > 0\) for all \(x\) in an interval, then the function is increasing on that interval because its derivative is positive, indicating that the function is "going up" as \(x\) increases. - If \(f'(x) < 0\) for all \(x\) in an interval, then the function is decreasing on that interval because its derivative is negative, indicating that the function is "going down" as \(x\) increases. Therefore, by examining the sign of the first derivative over different intervals, we can determine where the function is increasing or decreasing. This information is crucial in understanding the behavior of the function and identifying its critical points (such as maxima, minima, or points of inflection). In summary, the first derivative test provides a powerful tool for analyzing the behavior of a function and identifying intervals of increasing or decreasing behavior based on the sign of the derivative.