What You'll Master
This chapter teaches you how to use derivatives as a tool for understanding function behaviour and solving real-world optimization problems. You'll learn to identify where functions are climbing or falling, pinpoint their highest and lowest points, and apply these ideas to practical scenarios like maximizing area or minimizing cost.
Increasing & Decreasing
Use f'(x) to determine exactly where a function goes up and where it goes down.
Stationary Points
Find points where the gradient is zero, then classify them as maxima, minima, or inflection points.
Sketching Derivatives
Draw the graph of f'(x) by reading the original function f(x).
Optimization
Use calculus to find the maximum or minimum values in applied problems.
1. Increasing & Decreasing Functions
The Core Idea
When you look at a graph, you can see where it's going uphill (increasing) and where it's going downhill (decreasing). Your derivative tells you exactly the same thing — but with mathematical precision.
The derivative f'(x) is the gradient (slope) of the tangent line at any point. When f'(x) is positive, the tangent is tilted upwards, so the function is increasing. When f'(x) is negative, the tangent is tilted downwards, so the function is decreasing.
Definition
A function f(x) is increasing on an interval [a, b] if f'(x) ≥ 0 for all x in (a, b).
If f'(x) > 0 for all x in the interval, we say it is strictly increasing — the graph never plateaus.
Definition
A function f(x) is decreasing on an interval [a, b] if f'(x) ≤ 0 for all x in (a, b).
If f'(x) < 0 for all x in the interval, we say it is strictly decreasing — the graph never plateaus.
How to Find Increasing & Decreasing Intervals
Step 1: Find the first derivative f'(x).
Step 2: Solve the inequality f'(x) > 0 to find where the function is increasing.
Step 3: Solve the inequality f'(x) < 0 to find where the function is decreasing.
The boundary points (where f'(x) = 0) are called stationary points. These mark the transition between increasing and decreasing regions.
Worked Example
Question: For f(x) = x³ + x² − x + 2, find the intervals where f(x) is increasing and decreasing.
Step 1: Find f'(x)
f'(x) = 3x² + 2x − 1
Step 2: Find where f'(x) = 0
3x² + 2x − 1 = 0
(3x − 1)(x + 1) = 0
x = 1/3 or x = −1
Step 3: Test the sign of f'(x) in each region
• For x < −1, try x = −2:
f'(−2) = 3(4) + 2(−2) − 1 = 12 − 4 − 1 = 7 > 0 ✓
• For −1 < x < 1/3, try x = 0:
f'(0) = −1 < 0 ✗
• For x > 1/3, try x = 1:
f'(1) = 3 + 2 − 1 = 4 > 0 ✓
Answer:
f(x) is INCREASING on (−∞, −1) and (1/3, ∞)
f(x) is DECREASING on (−1, 1/3)
Note: You can use either strict inequalities (> and <) or non-strict ones (≥ and ≤) when describing intervals. Both are acceptable in exam answers.
Practice Question 1
For f(x) = 2x³ − 9x² + 12x − 1, find the intervals on which f is increasing and decreasing.
2. Stationary Points & Turning Points
What is a Stationary Point?
stationary point
At a stationary point: f'(x) = 0
How to Find Stationary Points
Step 1: Find the first derivative f'(x).
Step 2: