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

Verified step by step guidance

1. <Step 1: Find the derivative of the function.> Calculate the derivative of the function \( f(x) = x^2 - 4x + 3 \) to find \( f'(x) \).
2. <Step 2: Set the derivative equal to zero.> Solve \( f'(x) = 0 \) to find the critical points, which are potential points where the function changes from increasing to decreasing or vice versa.
3. <Step 3: Determine the critical points.> Solve the equation from Step 2 to find the values of \( x \) that are critical points.
4. <Step 4: Test intervals around the critical points.> Use test points in the intervals determined by the critical points to see where \( f'(x) > 0 \) (function is increasing) and where \( f'(x) < 0 \) (function is decreasing).
5. <Step 5: Identify the largest open intervals.> Based on the test results, identify the largest open interval where the function is increasing and the largest open interval where it is decreasing.

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.