Increasing and Decreasing Functions- First Derivative Test Explained

What the Hell Are Increasing and Decreasing Functions?

Let's cut through the textbook nonsense. A function is increasing when its output goes up as you move left to right along the x-axis. A function is decreasing when its output goes down as x increases.

That's it. That's the whole idea.

The derivative tells you the slope of the function at any point. When the derivative is positive, the function climbs upward. When it's negative, the function drops. When it's zero, you hit a potential turning point.

The First Derivative Test: What It Actually Does

The First Derivative Test is a method for finding relative maxima and minima of a function. It tells you where the function changes direction—from climbing to falling (local max) or from falling to climbing (local min).

You don't guess. You calculate.

How It Works

You find the critical points first—where f'(x) = 0 or where f'(x) doesn't exist. Then you check what happens to the derivative on either side of each critical point.

Step-by-Step: Applying the Test

Example: f(x) = x³ - 3x² - 9x + 5

Step 1: Find the derivative

f'(x) = 3x² - 6x - 9

Step 2: Set f'(x) = 0 and solve

3x² - 6x - 9 = 0
Divide by 3: x² - 2x - 3 = 0
Factor: (x - 3)(x + 1) = 0

Critical points: x = -1 and x = 3

Step 3: Test intervals

Pick test values in each interval created by your critical points.

Step 4: Read the results

At x = -1: derivative changes from positive to negative → local maximum
At x = 3: derivative changes from negative to positive → local minimum

Quick Reference Table

Sign Change of f'(x) What It Means Type of Point
+ to − Climbing then falling Local Maximum
− to + Falling then climbing Local Minimum
+ to + Keeps climbing No extremum
− to − Keeps falling No extremum

Where Students Screw Up

Forgetting to check endpoints: The First Derivative Test only finds relative maxima and minima. It won't tell you absolute maximum or minimum values on a closed interval. For those, you check endpoints separately.

Assuming f'(x) = 0 means a max or min: It doesn't. A flat spot doesn't guarantee a turning point. Check the sign change. If f'(x) = 0 but the derivative stays positive on both sides, you have a saddle point or plateau, not an extremum.

Only testing one side: You need both sides. A critical point with positive derivative on the left and positive derivative on the right is still just a flat spot on a rising function.

Ignoring where f'(x) doesn't exist: Critical points occur where the derivative is zero OR undefined. Vertical tangents, cusps, and corners count. Don't skip them.

When to Use the First Derivative Test

Use this test when you need to find the exact location of maxima and minima and you have a differentiable function. It's cleaner than the Second Derivative Test when the algebra for the second derivative gets messy.

The Second Derivative Test fails when f''(x) = 0 at your critical point. The First Derivative Test doesn't care about that—it only cares about direction change.

The Bottom Line

Find where f'(x) = 0. Check the sign on both sides. The direction change tells you what you need. That's the entire First Derivative Test.

No memorization tricks. No complicated formulas. Just calculate, test, and conclude.