Graphical Applications of Derivatives: Increasing/Decreasing Functions and Local Extrema

Graphical Applications of Derivatives

Increasing, Decreasing, and Constant Functions

The behavior of a function—whether it is increasing, decreasing, or constant—can be determined by examining the sign of its derivative. This is fundamental in calculus for understanding how functions change over intervals.

• Increasing Function: If $f'(x) > 0$ for all $x$ in an interval, then $f(x)$ is increasing on that interval.

• Decreasing Function: If $f'(x) < 0$ for all $x$ in an interval, then $f(x)$ is decreasing on that interval.

• Constant Function: If $f'(x) = 0$ for all $x$ in an interval, then $f(x)$ is constant on that interval.

Example: For $f(x) = 4x^3 + 3x^2 - 18x + 1$, compute $f'(x)$ and analyze its sign to determine intervals of increase and decrease.

Maxima and Minima (Local Extrema)

A function may have points where it reaches a local maximum or minimum, known as local extrema. These points are important for understanding the function's overall shape and behavior.

• Local Maximum: $f$ has a local maximum at $c$ if $f(x) \leq f(c)$ for all $x$ near $c$.

• Local Minimum: $f$ has a local minimum at $c$ if $f(x) \geq f(c)$ for all $x$ near $c$.

• Local Extremum: A point $c$ where $f$ has either a local maximum or minimum.

Application: Identifying local extrema is crucial in optimization problems and in analyzing real-world data, such as business trends.

Increasing/Decreasing Tests

To determine where a function is increasing or decreasing, follow these steps:

1. Compute the derivative: Find $f'(x)$.

2. Find critical numbers: Solve $f'(x) = 0$ or where $f'(x)$ is undefined.

3. Test intervals: Use the critical numbers to divide the domain into intervals. Test a value in each interval to determine the sign of $f'(x)$.

The intervals where $f'(x) > 0$ are where $f$ is increasing; intervals where $f'(x) < 0$ are where $f$ is decreasing.

Critical Numbers and Local Extrema

A critical number of a function $f$ is a value $c$ in the domain where $f'(c) = 0$ or $f'(c)$ does not exist. Local extrema occur only at critical numbers.

• Example: Find the critical numbers for $f(x) = \frac{1}{3}x^3 - x^2 - 15x + 6$ and $f(x) = \frac{1}{x-1}$ by solving $f'(x) = 0$ or where $f'(x)$ is undefined.

First Derivative Test

The First Derivative Test is used to classify critical points as local maxima, minima, or neither, based on the sign of the derivative before and after the critical point.

• If $f'(a) > 0$ and $f'(b) < 0$, then there is a local maximum at $c$.

• If $f'(a) < 0$ and $f'(b) > 0$, then there is a local minimum at $c$.

• If $f'(a)$ and $f'(b)$ are both positive or both negative, then there is no local extremum at $c$.

Example: Use the first derivative test to find the location of all extrema for $f(x) = x^3 + 4x^2 - 3x + 5$ and $f(x) = \frac{1}{3}x^3 - x^2 - 8x + 57$.

Business Application: Unemployment Rate Analysis

Graphical analysis of real-world data, such as the unemployment rate, can be performed using calculus concepts. By examining the graph and its slopes, we can answer questions about trends and identify local minima and maxima.

• At the start of 2008: Estimate the unemployment rate from the graph.

• During 2008: Determine if the rate was increasing or decreasing by observing the slope.

• Local Minimum: Identify the year where the unemployment rate reached a local minimum.

• Local Maximum: Identify the year where the unemployment rate reached a local maximum.

Additional info: These applications demonstrate how calculus is used to interpret and analyze data in economics and business, providing insight into trends and turning points.