To determine whether a function is increasing or decreasing, we look at its derivative. The derivative of a function gives us the slope of the tangent line at any point on the graph of the function.
If the derivative of a function, denoted as f'(x), is positive over an interval, this means that the function f(x) is increasing on that interval. In other words, as x increases, the value of f(x) also increases.
Conversely, if the derivative f'(x) is negative on an interval, then the function f(x) is decreasing on that interval. This indicates that as x increases, the value of f(x) decreases.
To summarize:
- If f'(x) > 0, then f(x) is increasing.
- If f'(x) < 0, then f(x) is decreasing.
It’s also helpful to consider points where the derivative is zero, known as critical points. At these points, the function might change from increasing to decreasing or vice versa. Therefore, analyzing the sign of the derivative around these critical points helps determine the behavior of the function more thoroughly.