IB Maths AI HLDifferentiationPaper 1 & 2~7 min read
Increasing & Decreasing Functions
A gradient at a single point told you the steepness there; its sign tells you the direction of travel. Where f′(x) > 0 the curve climbs (increasing), where f′(x) < 0 it falls (decreasing), and where f′(x) = 0 it’s momentarily flat (stationary). Finding intervals is just solving an inequality in the derivative.
📘 What you need to know
Increasing: f′(x) > 0 — the output rises as x rises.
Decreasing: f′(x) < 0 — the output falls as x rises.
Stationary: f′(x) = 0 — momentarily flat (the dividing points).
At a point: substitute the x-value into f′(x) and read the sign.
Over an interval: solve the inequality f′(x) > 0 (or < 0) to get a range of x.
Answer as an interval: give a range of x, not a single value.
Sign of the derivative = direction
The whole idea fits on one line: the sign of f′(x) tells you whether the curve is heading uphill, downhill, or sitting flat at that x.
increasingf′(x) > 0Positive gradient — the curve climbs as x increases.
decreasingf′(x) < 0Negative gradient — the curve falls as x increases.
Stationary (the boundary)f′(x) = 0
The dividing point between increasing and decreasing ✗ not in booklet — it’s a definition
To go from climbing to falling, the curve has to stop rising and start dropping — and the only way to switch from positive to negative gradient is to pass through zero. That’s why every local max or min sits exactly where f′(x) = 0, with increasing on one side and decreasing on the other. (The reverse signals a minimum.)
Finding the intervals
Most curves are a patchwork of increasing bits, decreasing bits, and stationary points in between. To pin down where each happens, differentiate and solve the matching inequality.
🧭 Recipe — increasing / decreasing intervals
Differentiate to get f′(x).
At a point: substitute the x-value and check the sign (+ = increasing, − = decreasing).
Over an interval: solve f′(x) > 0 for increasing, or f′(x) < 0 for decreasing.
State the range of x (e.g. x > ½), and note any stationary points where f′(x) = 0.
Worked illustration: for f(x) = x2, f′(x) = 2x, so it’s decreasing for x < 0, stationary at x = 0, and increasing for x > 0 — exactly the U-shape you’d expect.
🧠 “Plus climbs, minus dives”
A plus gradient means the graph climbs; a minus gradient means it dives. Zero is the level pause in between. Read the sign of f′, not f.
Worked examples
All use the SME function f(x) = x2 − x − 2.
WE 1
Find f′(x)
Differentiate to get the gradient function.
f(x) = x² − x − 2f′(x) = 2x − 1
WE 2
Increasing or decreasing at x = 0?
Substitute x = 0 into the derivative and read the sign.
f′(0) = 2(0) − 1 = −1−1 < 0decreasing at x = 0
WE 3
Increasing or decreasing at x = 3?
Same idea at x = 3.
f′(3) = 2(3) − 1 = 55 > 0increasing at x = 3
WE 4
For which x is f increasing?
Increasing means f′(x) > 0 — solve the inequality.
f′(x) > 0 → 2x − 1 > 02x > 1 → x > ½increasing for x > ½
WE 5
For which x is f decreasing, and where is it stationary?
Mirror the previous step with the < 0 inequality, and find the boundary.
f′(x) < 0 → 2x − 1 < 0 → x < ½stationary: 2x − 1 = 0 → x = ½decreasing for x < ½, stationary at x = ½x = ½ is the turning point of the parabola (a minimum).
💡 Top tips
Read the sign of f′(x), not f(x) — direction comes from the derivative.
At a point: substitute and check the sign. Over an interval: solve the inequality.
Answer with a range of x (an interval), not a single number.
Stationary points are the boundaries between increasing and decreasing — set f′(x) = 0.
Most curves mix all three — increasing, decreasing, and stationary in different regions.
Check direction with the graph on your GDC if unsure of a sign.
⚠ Common mistakes
Using f(x) instead of f′(x) — increasing/decreasing is about the gradient’s sign.
Giving a single value when the answer is an interval (a range of x).
Flipping the inequality wrong — increasing is f′(x) > 0, decreasing is < 0.
Forgetting the stationary points sit exactly where f′(x) = 0.
Sign slip on substitution — double-check arithmetic for negative x-values.
Next up — Local Minimum & Maximum Points. You’ve now seen that the sign of f′(x) flips at a turning point, passing through zero. The next topic zooms in on exactly those f′(x) = 0 points: how to locate them, find their coordinates, and decide their nature — whether the curve changes from decreasing-to-increasing (a local minimum) or increasing-to-decreasing (a local maximum).
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