How do you find the increasing and decreasing intervals of a fraction?

Here’s a revised version of the blog post, aiming for a more human and engaging tone:

Unlocking Function Behavior: Finding Where Things Go Up and Down

Ever wondered how to really see what a function is doing? Beyond just plugging in numbers, I mean? One of the coolest tricks in calculus is figuring out where a function is increasing (going uphill, so to speak) and decreasing (heading downhill). Trust me, this isn’t just some abstract math concept. It’s a powerful tool for sketching graphs, solving tricky optimization problems, and even understanding how things change in the real world. So, let’s dive in!

First, Let’s Get on the Same Page

Okay, so what do “increasing” and “decreasing” actually mean in math terms? Think of it like this:

• Increasing Function: Imagine you’re walking along the graph from left to right. If you’re always going uphill, that’s an increasing function. Mathematically, if you pick any two points where x₁ is smaller than x₂, then f(x₁) will also be smaller than f(x₂). Simple as that!
• Decreasing Function: You guessed it! This is when you’re walking downhill. As x gets bigger, f(x) gets smaller.
• Critical Points: The Turning Points: Now, these are super important. Critical points are those spots where the function’s derivative (f'(x)) is either zero or doesn’t exist. Think of them as potential peaks and valleys – the places where the function might switch from going up to going down, or vice versa. They’re the key to unlocking the whole puzzle.

Time to Get Our Hands Dirty: Finding the Derivative

Alright, first things first: we need to find the derivative of our function, f'(x). Remember, the derivative tells us the instantaneous rate of change – how quickly the function is changing at any given point. Depending on the function, you might need to use the power rule, product rule, quotient rule, chain rule… it’s like having a whole toolbox of differentiation techniques!

Hunting for Critical Points: The Detective Work

Now for the fun part: let’s find those critical points! Set f'(x) equal to zero and solve for x. These are your prime suspects. But don’t forget to check where f'(x) is undefined. Maybe there’s a sneaky division by zero lurking somewhere? Those points are critical too, and you definitely don’t want to miss them.

The Number Line: Our Investigation Board

Okay, grab a number line and mark all those critical points you just found. These points chop up the number line into different intervals, each representing a potential increasing or decreasing zone. Now, the real detective work begins. Pick a test value, a “spy,” from each interval and plug it into f'(x).

• If f'(c) is positive (greater than 0), that means the function is increasing in that interval. Go uphill!
• If f'(c) is negative (less than 0), the function is decreasing in that interval. Time to go downhill!
• And if f'(c) is zero? Well, that just tells us that at that specific point the function is momentarily flat. It doesn’t tell us anything about the whole interval.

Announcing Our Findings: The Increasing and Decreasing Intervals

Based on whether f'(x) is positive or negative in each interval, you can now confidently state where the function is increasing and decreasing. Use interval notation to present your findings. For instance, if f(x) is increasing before x = 2 and decreasing after that, you’d write:

• Increasing interval: (-∞, 2)
• Decreasing interval: (2, ∞)

Let’s See It in Action: An Example

Let’s crack a case together. Consider f(x) = x³ – 3x² + 1.