Function Increasing or Decreasing- Analyzing Behavior
What "Increasing" and "Decreasing" Actually Mean
Before we get into the math, let's be clear about what these terms represent. A function is increasing on an interval when, as you move right along the x-axis, the function values go up. A function is decreasing when the function values go down as x increases.
That's it. No fancy definitions. Just movement direction.
Graphically, you can spot this by asking: if you trace the curve from left to right, does it go up or down? 👆📈
The Derivative Test: Your Primary Tool
The first derivative tells you the slope of a function at any point. This is how you determine increasing/decreasing behavior.
The Rules
- If f'(x) > 0 for all x in an interval → the function is increasing there
- If f'(x) < 0 for all x in an interval → the function is decreasing there
- If f'(x) = 0 → you have a critical point (could be a max, min, or saddle)
Positive derivative means climbing. Negative derivative means falling. Zero derivative means you're at a peak, valley, or flat spot.
Finding Critical Points First
You can't determine behavior without finding where the derivative equals zero or doesn't exist. These are your critical points.
To find them:
- Take the derivative f'(x)
- Set f'(x) = 0 and solve
- Find where f'(x) does not exist
- Plot these x-values on a number line
These points divide your domain into intervals. Test each interval separately.
The First Derivative Test: Step-by-Step
Here's how to actually analyze a function's behavior:
Step 1: Find f'(x)
Take the derivative of your function. Work it out completely.
Step 2: Find Critical Numbers
Solve f'(x) = 0. Factor if possible. These are your boundary candidates.
Step 3: Test the Intervals
Pick one test point in each interval created by your critical numbers. Plug into f'(x).
- Positive result → increasing on that interval
- Negative result → decreasing on that interval
Step 4: Classify the Critical Points
Use the sign changes around each critical point:
- f' changes from positive to negative → local maximum
- f' changes from negative to positive → local minimum
- No sign change → neither max nor min (inflection or flat)
Working Example
Let's analyze f(x) = x³ - 3x² - 9x + 5
Step 1: Find the derivative
f'(x) = 3x² - 6x - 9
Step 2: Set equal to zero and solve
3x² - 6x - 9 = 0
Divide by 3: x² - 2x - 3 = 0
Factor: (x - 3)(x + 1) = 0
Critical points: x = 3 and x = -1
Step 3: Test intervals
Number line divided at x = -1 and x = 3
- Test x = -2: f'(-2) = 3(4) - 6(-2) - 9 = 12 + 12 - 9 = 15 > 0 → increasing
- Test x = 0: f'(0) = -9 < 0 → decreasing
- Test x = 4: f'(4) = 3(16) - 6(4) - 9 = 48 - 24 - 9 = 15 > 0 → increasing
Step 4: Classification
At x = -1: f' changes from positive to negative → local maximum
At x = 3: f' changes from negative to positive → local minimum
First vs. Second Derivative: When to Use Which
The first derivative tells you about increasing/decreasing. The second derivative tells you about concavity (curving up or down).
You need both for complete behavior analysis:
| What You Want to Know | Use This Derivative | Rule |
|---|---|---|
| Where function rises | First: f'(x) | f'(x) > 0 |
| Where function falls | First: f'(x) | f'(x) < 0 |
| Where curve bends upward | Second: f''(x) | f''(x) > 0 (concave up) |
| Where curve bends downward | Second: f''(x) | f''(x) < 0 (concave down) |
| Inflection points | Second: f''(x) | f''(x) = 0 or undefined |
Common Mistakes That Cost You Points
- Forgetting to test endpoints — critical points only tell you about the neighborhood, not the full interval
- Assuming f'(x) = 0 means maximum or minimum — it could be a saddle point (like x³ at x=0)
- Not checking where f'(x) doesn't exist — vertical tangents and cusps are also critical points
- Confusing "concave up" with "increasing" — a function can be increasing while concave down (like √x)
Quick Reference: Sign Pattern Analysis
| Sign Pattern of f'(x) | Function Behavior | Critical Point Type |
|---|---|---|
| +/− (positive to negative) | ↗ then ↘ | Local maximum |
| −/+ (negative to positive) | ↘ then ↗ | Local minimum |
| +/+ or −/− (no change) | Always rising or always falling | No local extremum |
Getting Started: Your Analysis Checklist
When you're given a function and asked to analyze increasing/decreasing behavior:
- Find f'(x) — take the derivative correctly
- Solve f'(x) = 0 — find all critical numbers
- Draw a number line — mark all critical numbers
- Pick test points — one in each interval between critical numbers
- Evaluate f'(x) at each test point — record positive or negative
- Mark behavior on the number line — + means increasing, − means decreasing
- Identify maxima and minima — look for sign changes
That's the complete process. No shortcuts that actually work.
Why This Matters Beyond Homework
Understanding function behavior isn't abstract math nonsense. It applies directly to:
- Optimization — finding maximum profit, minimum cost, optimal dimensions
- Physics — velocity and acceleration relationships (velocity is f', acceleration is f'')
- Economics — marginal cost, diminishing returns, growth rates
- Data analysis — identifying trends, peaks, and valleys in real data
The derivative tests are tools. You learn them here so you can use them everywhere else.
The Bottom Line
Increasing/decreasing analysis comes down to one thing: the sign of the first derivative. Find where it's positive, you have increase. Find where it's negative, you have decrease. Find where it changes sign, you have extrema.
Work the problems. Test intervals systematically. Don't skip steps to save time — that's how you get wrong answers.