Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = x^2 - 4x + 3

Key Concepts

Function Behavior

Understanding how a function behaves involves analyzing its increasing and decreasing intervals. A function is increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, it is decreasing if the function's value at the second point is less than at the first. This behavior is crucial for determining the intervals of increase and decrease.

Critical Points

Critical points are values of x where the derivative of the function is zero or undefined. These points are essential for identifying where a function changes from increasing to decreasing or vice versa. By finding the critical points of the function, we can analyze the intervals around these points to determine where the function is increasing or decreasing.

First Derivative Test

The First Derivative Test is a method used to determine the nature of critical points. By evaluating the sign of the derivative before and after a critical point, we can conclude whether the function is increasing or decreasing in the intervals surrounding that point. If the derivative changes from positive to negative, the function is increasing before the critical point and decreasing afterward, indicating a local maximum.