Core Definitions
↑ Strictly Increasing
↓ Strictly Decreasing
A function f is strictly increasing on (a,b) if for every x₁ < x₂ in (a,b), we have f(x₁) < f(x₂).
A function f is strictly decreasing on (a,b) if for every x₁ < x₂ in (a,b), we have f(x₁) > f(x₂).
The Derivative Test
f'(x) > 0 → Increasing | f'(x) < 0 → Decreasing | f'(x) = 0 → Stationary
Key Rules
- 1Find f'(x), then solve f'(x) = 0 to get critical points.
- 2Critical points divide the number line into intervals. Test the sign of f'(x) in each interval.
- 3If f'(x) > 0 on an interval → f is strictly increasing there.
- 4If f'(x) < 0 on an interval → f is strictly decreasing there.
- 5A function can be monotonic (always increasing or always decreasing) on its entire domain.
- 6Endpoints are included in closed intervals [ ] for increasing/decreasing intervals.
Tip: If f'(x) ≥ 0 for all x and f'(x) = 0 only at isolated points, the function is still strictly increasing.
Monotonic Functions
Monotonically Increasing / Decreasing
A function which is either always increasing or always decreasing throughout its domain is called monotonic. Example: f(x) = eˣ is monotonically increasing for all x ∈ ℝ.
Solved Examples
Example 01
Find intervals for f(x) = x² – 4x + 3
Q. Find the intervals in which f(x) = x² – 4x + 3 is increasing or decreasing.
f'(x) = 2x – 4
f'(x) = 0 → 2x – 4 = 0 → x = 2 (critical point)
For x < 2: f'(x) = 2(1) – 4 = –2 < 0 → Decreasing on (–∞, 2)
For x > 2: f'(x) = 2(3) – 4 = 2 > 0 → Increasing on (2, +∞)
∴ Decreasing on (–∞, 2] and Increasing on [2, +∞)
Example 02
Find intervals for f(x) = 2x³ – 9x² + 12x – 5
Q. Find increasing and decreasing intervals for f(x) = 2x³ – 9x² + 12x – 5.
f'(x) = 6x² – 18x + 12 = 6(x² – 3x + 2) = 6(x–1)(x–2)
f'(x) = 0 → x = 1 or x = 2
x < 1: f'(0) = 6(–1)(–2) = +12 > 0 → Increasing
1 < x < 2: f'(1.5) = 6(0.5)(–0.5) < 0 → Decreasing
x > 2: f'(3) = 6(2)(1) = +12 > 0 → Increasing
∴ Increasing: (–∞, 1] ∪ [2, ∞) | Decreasing: [1, 2]
Example 03
Show f(x) = x³ is strictly increasing
Q. Show that f(x) = x³ is strictly increasing on ℝ.
f'(x) = 3x²
3x² ≥ 0 for all x, and 3x² = 0 only at x = 0 (an isolated point)
∴ f(x) = x³ is strictly increasing on ℝ (monotonically increasing)