Question

How to find where f is increasing and decreasing?

Answer 1

To find where a function f(x) is increasing or decreasing, calculate its derivative f'(x) and analyze the sign over different intervals. The function is increasing where f'(x) is positive and decreasing where f'(x) is negative. Critical points, where f'(x) changes sign, help identify transitions between increasing and decreasing intervals.

Step-by-step explanation:

To determine where the function f(x) is increasing or decreasing, we need to look at the slope of the function at various points. If the slope is positive, then the function is increasing; if the slope is negative, the function is decreasing.

Identifying Increments and Decrements in Functions

Firstly, consider the derivative f'(x), since it represents the slope of f(x) at any given point. If f'(x) is positive for an interval, f(x) is increasing on that interval. Conversely, if f'(x) is negative, f(x) is decreasing. The points where f'(x) changes sign are called critical points and can be local maxima, minima, or points of inflection.

For instance, referring to the discussion about the product of f and λ being constant, one can infer indirectly about the behavior of f when λ changes: as f becomes smaller, λ must increase, implying a negative slope at points where f is decreasing and vice versa.

To put this into practice, consider a function with different parts described in the information provided:

• Part A has a nonzero y-intercept and initially decreases.
• Part B has a zero starting point and is increasing.
• Part C again starts at zero but the rate of increase changes before it levels off.

The description of the part with the nonzero y-intercept and downward slope that levels off, suggests a decreasing function initially, until it levels off, at which point it stops decreasing. The parts described as beginning at zero with an upward slope, indicate sections of the function that are increasing.

To analyze this methodically:

1. Find the derivative f'(x).
2. Determine the sign of f'(x) over different intervals.
3. Identify the critical points where f'(x) changes sign from positive to negative or vice versa.
4. Use this information to determine where the function is increasing or decreasing.