IB Maths AA SL Topic 5 — Calculus Paper 1 & 2 ~9 min read

Increasing & Decreasing Functions

Look at any graph: in some places it goes up, in some it goes down, and at certain points it’s perfectly flat. The clever bit? You don’t need a graph to figure out which is which — you just look at the sign of the derivative. Positive means the curve is rising, negative means it’s falling, zero means it’s stationary. Same idea, three little signs.

📘 What you need to know

A function f(x) is increasing when f′(x) > 0 → output goes up as x goes up.
A function f(x) is decreasing when f′(x) < 0 → output goes down as x goes up.
A function is stationary where f′(x) = 0 → curve is momentarily flat.
To find where a function is increasing or decreasing, find f′(x) and solve the inequalities f′(x) > 0 or f′(x) < 0.
To check at a single point, substitute the x-value into f′(x) and look at the sign of the answer.
Most curves are a mix — they have intervals where they increase, intervals where they decrease, and stationary points where they switch direction.

What does “increasing” or “decreasing” actually mean?

It’s the simplest idea in calculus, but worth nailing down precisely. As you move along the graph from left to right (i.e. as x gets bigger):

If the output (y-value) goes up, the function is increasing.
If the output goes down, it’s decreasing.
If the output isn’t changing at all (the curve is flat for an instant), it’s stationary.

Don’t confuse “the function’s value is positive” with “the function is increasing”. They’re different things! A function can have a positive value while still going downhill — and vice versa. Always think about the direction, not the height.

The sign of the derivative tells you everything

Here’s the rule in one line: the sign of f′(x) tells you which of the three states the function is in. That’s the entire topic in one move.

Increasing

f′(x) > 0

positive gradient — output rises

Stationary

f′(x) = 0

flat — momentarily not changing

Decreasing

f′(x) < 0

negative gradient — output falls

🧠

“Plus up, minus down, zero flat”

Three signs, three behaviours. Plus → goes up. Minus → comes down. Zero → flat. Once you’ve drilled this for ten seconds, you’ll never forget which sign means which.

Putting it all together on one curve

A typical curve has all three states at different places. Look at this cubic — it’s increasing, then decreasing, then increasing again, with two stationary points where it switches direction:

A cubic showing all three states

🤔 What’s special about the stationary points?

Stationary points are where the function switches direction — going from increasing to decreasing, or vice versa. At those points, the gradient is exactly zero (the curve is flat for an instant). They’re how you spot maximums and minimums. We’ll come back to those in detail later in calculus.

Checking the behaviour at a single point

If a question asks “is f(x) increasing or decreasing at x = 3?“, you don’t need to find a whole interval — just plug the value into f′(x) and look at the sign.

If f′(3) is positive, the function is increasing at x = 3.
If f′(3) is negative, the function is decreasing at x = 3.
If f′(3) = 0, it’s stationary at x = 3.

📍

Don’t compute the value — just check the sign

You don’t even need an exact number. If you can tell whether f′(3) is positive or negative, that’s enough. State the sign, then state the conclusion: “f′(3) = 5 > 0, so f is increasing at x = 3.”

Finding the intervals where it’s increasing or decreasing

If a question asks for the range of x-values where the function is increasing (or decreasing), you need to solve an inequality:

Where is f(x) increasing or decreasing?

Increasing: solve f′(x) > 0

Decreasing: solve f′(x) < 0

The 2-step method

Differentiate to find f′(x).
Solve the inequality for the type of interval the question asks for (increasing → > 0, decreasing → < 0).

Sign diagrams: a useful visual tool

For more complex derivatives — quadratics or cubics — it helps to draw a quick number line marking where f′(x) is zero, and then test which intervals give positive or negative values. Here’s an example for f′(x) = 2x − 1, where the only root is x = ½:

Sign of f′(x) = 2x − 1

−

decreasing

x = ½

f′ = 0

increasing

From the sign diagram you can read off both intervals at a glance: f is decreasing for x < ½ and increasing for x > ½.

For a basic linear derivative like 2x − 1, you can probably solve the inequality directly. But for quadratic or cubic derivatives, sign diagrams save time and stop you from missing intervals. Pick a test point in each region (e.g. x = 0 and x = 1) and check the sign — that tells you which side is positive and which is negative.

Worked examples

WE 1

Check increasing/decreasing at single points

For f(x) = x² − x − 2, determine whether f(x) is increasing or decreasing at the points where x = 0 and x = 3.

Differentiate, then check the sign of f′(x) at each point.step 1 — differentiate f′(x) = 2x − 1at x = 0 f′(0) = 2(0) − 1 = −1 < 0 Negative → decreasing. at x = 0, f(x) is decreasingat x = 3 f′(3) = 2(3) − 1 = 5 > 0 Positive → increasing. at x = 3, f(x) is increasing always state the sign first, then the conclusion — earns method marks!

WE 2

Find the interval where a function is increasing

For f(x) = x² − x − 2 (same function as WE 1), find the values of x for which f(x) is increasing.

f(x) is increasing when f′(x) > 0. Solve the inequality. From WE 1: f′(x) = 2x − 1 Set up: 2x − 1 > 0 2x > 1 → x > ½ f(x) is increasing for x > ½ just solve the inequality like any other algebra question — the calculus is over after step 1!

WE 3

Find where a cubic is decreasing

Find the values of x for which f(x) = x³ − 6x² + 9x + 1 is decreasing.

Differentiate, factorise the quadratic derivative, then solve f′(x) < 0.step 1 — differentiate f′(x) = 3x² − 12x + 9step 2 — factorise f′(x) = 3(x² − 4x + 3) = 3(x − 1)(x − 3)step 3 — solve f′(x) < 0 3(x − 1)(x − 3) is negative between the roots: 1 < x < 3 f(x) is decreasing for 1 < x < 3 a positive quadratic with roots a, b is negative between them — handy shortcut!

WE 4

Multi-part: stationary, increasing, decreasing

For g(x) = x³ − 3x² + 4:

(a) Find the x-coordinates of the stationary points.
(b) Find the values of x for which g(x) is increasing.

Stationary → g′(x) = 0. Increasing → g′(x) > 0.setup — differentiate g′(x) = 3x² − 6x = 3x(x − 2)part (a) — stationary points 3x(x − 2) = 0 → x = 0 or x = 2 stationary at x = 0 and x = 2part (b) — increasing intervals 3x(x − 2) > 0 (positive quadratic, roots 0 and 2): positive OUTSIDE the roots: x < 0 OR x > 2 g(x) is increasing for x < 0 or x > 2 use “or” for separated intervals — never write “0 > x > 2” backwards!

WE 5

Show a function is always increasing

Show that h(x) = x³ + 3x + 5 is an increasing function for all real x.

“Always increasing” means h′(x) > 0 for every x. Show that h′(x) is always positive. Differentiate: h′(x) = 3x² + 3 Notice 3x² ≥ 0 for all x (a square is never negative). So: h′(x) = 3x² + 3 ≥ 0 + 3 = 3 > 0 Therefore h′(x) > 0 for all real x. h(x) is increasing for all x ∈ ℝ ✓ “squared term + positive constant” is ALWAYS positive — useful “show that” trick!

💡 Top tips

The sign of f′(x) is everything. Positive → increasing. Negative → decreasing. Zero → stationary.
For a single-point check, just substitute and report the sign — you don’t need to know the actual gradient value.
For an interval, set up an inequality with f′(x) and solve like normal algebra.
Factorise the derivative when it’s a quadratic — it makes the inequality much easier to read.
Use a sign diagram for cubic or higher derivatives. Test one point in each region to find the sign.
Positive quadratic factorised: negative between the roots, positive outside. Memorise this.
For “show that” questions, rearrange f′(x) into a form that’s clearly always positive (or negative).
Always state your conclusion in words — “f is increasing for x > 2” — even after a calculation, examiners want the verbal answer.

⚠ Common mistakes

Using f(x) instead of f′(x). “Increasing” is about the gradient — you need the derivative, not the original function.
Confusing positive/negative function values with positive/negative gradients. f(x) can be negative while still increasing, and vice versa.
Saying “increasing for x > 0” when it’s a quadratic. Quadratics often have two intervals — don’t miss the second one.
Writing the inequality backwards. Use “or” for two separate intervals, not impossible chains like “0 > x > 2”.
Forgetting strict vs. non-strict inequalities. Some questions specify “strictly increasing” (use >) vs “increasing” (≥). Read carefully.
Sign errors when factorising. (x − 1)(x − 3) is negative between 1 and 3, not outside them.
Skipping the conclusion after the algebra. Always state in words which intervals you found.
Including stationary points in increasing/decreasing intervals. f′(x) = 0 means stationary, not increasing or decreasing — use strict > and < for those.

🎉 You’ve finished the introduction to differentiation! You can now find derivatives, work out gradients at any point, write tangent and normal equations, and identify where a function is rising, falling, or flat. From here, calculus opens up — finding maximum and minimum points, optimising real-world problems, and eventually integration (the reverse process). The foundations you’ve built here carry you through all of it.

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