Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = (x - 1)²

Verified step by step guidance

1. To determine where the function \( f(x) = (x - 1)^2 \) is increasing or decreasing, we first need to find its derivative, \( f'(x) \). Use the power rule to differentiate: \( f'(x) = 2(x - 1) \).
2. Set the derivative \( f'(x) = 2(x - 1) \) equal to zero to find the critical points. Solve \( 2(x - 1) = 0 \) to get \( x = 1 \). This is the critical point where the function could change from increasing to decreasing or vice versa.
3. Determine the sign of \( f'(x) \) on the intervals around the critical point. Choose a test point from each interval: for \( x < 1 \), choose \( x = 0 \); for \( x > 1 \), choose \( x = 2 \).
4. Evaluate \( f'(x) \) at the test points: \( f'(0) = 2(0 - 1) = -2 \) (negative, so \( f(x) \) is decreasing on \( (-\infty, 1) \)); \( f'(2) = 2(2 - 1) = 2 \) (positive, so \( f(x) \) is increasing on \( (1, \infty) \)).
5. Conclude that the function \( f(x) = (x - 1)^2 \) is decreasing on the interval \( (-\infty, 1) \) and increasing on the interval \( (1, \infty) \).

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus used to determine the slope of the tangent line to the curve at any point. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for analyzing the behavior of the function, as they can indicate potential local maxima, minima, or points of inflection. To determine intervals of increase or decrease, one must evaluate the derivative at these critical points and test the sign of the derivative in the intervals created by these points.

Test Intervals

Test intervals are segments of the domain of a function that are determined by critical points. By selecting test points within these intervals and evaluating the sign of the derivative, one can ascertain whether the function is increasing or decreasing in each interval. This method provides a systematic approach to understanding the overall behavior of the function across its domain.