# Increasing and Decreasing Functions

### Increasing and Decreasing Functions:

• Increasing Function: A function $f\left(x\right)$ is increasing on an interval if, as $x$ increases, the corresponding values of $f\left(x\right)$ also increase.
• Decreasing Function: A function $f\left(x\right)$ is decreasing on an interval if, as $x$ increases, the corresponding values of $f\left(x\right)$ decrease.

### Using Derivative Rules for Analysis:

1. Increasing Functions:

• For a function $f\left(x\right)$:
• ${f}^{\mathrm{\prime }}\left(x\right)>0$ on an interval $I$ implies $f\left(x\right)$ is increasing on $I$.
• Example: $f\left(x\right)={x}^{2}$:
• Calculate ${f}^{\mathrm{\prime }}\left(x\right)=2x$.
• ${f}^{\mathrm{\prime }}\left(x\right)>0$ for $x>0$, indicating $f\left(x\right)$ is increasing on $x>0$.
2. Decreasing Functions:

• For a function $f\left(x\right)$:
• ${f}^{\mathrm{\prime }}\left(x\right)<0$ on an interval $I$ implies $f\left(x\right)$ is decreasing on $I$.
• Example: $f\left(x\right)=-{x}^{3}$:
• Calculate ${f}^{\mathrm{\prime }}\left(x\right)=-3{x}^{2}$.
• ${f}^{\mathrm{\prime }}\left(x\right)<0$ for $x\mathrm{\ne }0$, indicating $f\left(x\right)$ is decreasing on $x\mathrm{\ne }0$.

### Using Derivative Rules for Increasing/Decreasing Functions:

1. Increasing Functions:

• If ${f}^{\mathrm{\prime }}\left(x\right)>0$ for all $x$ in an interval, then $f\left(x\right)$ is increasing on that interval.
• If ${f}^{\mathrm{\prime }}\left(x\right)$ is non-negative (or ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)\ge 0$) for all $x$ in an interval, then $f\left(x\right)$ is increasing on that interval.
2. Decreasing Functions:

• If ${f}^{\mathrm{\prime }}\left(x\right)<0$ for all $x$ in an interval, then $f\left(x\right)$ is decreasing on that interval.
• If ${f}^{\mathrm{\prime }}\left(x\right)$ is non-positive (or ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)\le 0$) for all $x$ in an interval, then $f\left(x\right)$ is decreasing on that interval.

### Example:

#### Example 1: Determining Increasing/Decreasing Intervals

Given $f\left(x\right)={x}^{3}-3{x}^{2}-9x+5$:

1. Find where $f\left(x\right)$ is increasing or decreasing.

Solution:

1. Derivative Calculation:

• Calculate ${f}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left({x}^{3}-3{x}^{2}-9x+5\right)=3{x}^{2}-6x-9$
2. Identifying Increasing/Decreasing Intervals:

• Setting ${f}^{\mathrm{\prime }}\left(x\right)>0$ for increasing intervals: $3{x}^{2}-6x-9>0$ ${x}^{2}-2x-3>0$ $\left(x-3\right)\left(x+1\right)>0$
• This inequality holds for $x<-1$ or $x>3$, indicating $f\left(x\right)$ is increasing on $x<-1$ or $x>3$.
• Setting ${f}^{\mathrm{\prime }}\left(x\right)<0$ for decreasing intervals: $3{x}^{2}-6x-9<0$ ${x}^{2}-2x-3<0$ $\left(x-3\right)\left(x+1\right)<0$
• This inequality holds for $-1, indicating $f\left(x\right)$ is decreasing on $-1.

### Strategies:

• Derivative Sign Analysis: Understand the relationship between the sign of the derivative and the behavior of the function (increasing or decreasing) on specific intervals.
• Interval Identification: Use the derivative sign to determine intervals where the function exhibits increasing or decreasing behavior.

#### Example 1: Determining Increasing/Decreasing Behavior

Given $f\left(x\right)={x}^{2}-2x$:

1. Find where $f\left(x\right)$ is increasing or decreasing.

Solution:

1. Derivative Calculation:
• Find ${f}^{\mathrm{\prime }}\left(x\right)$ to determine increasing/decreasing behavior: ${f}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left({x}^{2}-2x\right)=2x-2$
• Setting ${f}^{\mathrm{\prime }}\left(x\right)=0$ to find critical points: $2x-2=0$ $x=1$
• Test intervals around the critical point $x=1$:
• Test $x=0$ (left of $x=1$): ${f}^{\mathrm{\prime }}\left(0\right)=2\left(0\right)-2=-2\phantom{\rule{1em}{0ex}}\text{(Decreasing)}$
• Test $x=2$ (right of $x=1$): ${f}^{\mathrm{\prime }}\left(2\right)=2\left(2\right)-2=2\phantom{\rule{1em}{0ex}}\text{(Increasing)}$

### Strategies:

• Derivative Analysis: Use the derivative ${f}^{\mathrm{\prime }}\left(x\right)$ to determine the function's increasing or decreasing behavior.
• Critical Points: Identify critical points by setting ${f}^{\mathrm{\prime }}\left(x\right)$ equal to zero or where it is undefined to find potential points of change in behavior.
• Test Intervals: Test intervals around critical points by selecting test points and evaluating ${f}^{\mathrm{\prime }}\left(x\right)$ to determine the function's behavior within those intervals.