IB Maths AA HLTopic 5 — CalculusPaper 1 & 2~7 min read
Increasing & Decreasing Functions
The sign of f′(x) tells you whether f is increasing (f′ > 0), decreasing (f′ < 0) or stationary (f′ = 0). Solving the inequalities partitions the domain into intervals where the function rises, falls, or pauses. This is a fast way to map the shape of a curve without sketching it term by term.
📘 What you need to know
Increasing: f is increasing on an interval where f′(x) > 0.
Decreasing: f is decreasing on an interval where f′(x) < 0.
Stationary: f is stationary at points where f′(x) = 0.
At a single point: just evaluate f′(x₁) and check sign.
Over an interval: solve f′(x) > 0 (or < 0) — the answer is a range of x-values.
Polynomial f′ with multiple roots: factor, find sign in each region between roots.
“Always increasing”: show f′(x) > 0 for ALL real x — often via sum of non-negatives.
Strict vs non-strict: typically use strict (>, <) for monotonic regions and include = only for the boundary stationary points.
Where f′ > 0 the curve climbs; where f′ < 0 it falls; where f′ = 0 it pauses. The pause-points (stationary points) are where the regions switch.
Single critical point — solve a linear inequality
For a quadratic f, the derivative f′ is linear. Solving f′(x) > 0 gives a single half-line. The answer is one inequality.
Quick check: for f(x) = ax² + bx + c, f is increasing on the right side of the vertex if a > 0 (parabola opens upward) and on the left side if a < 0. The vertex itself is at x = −b/(2a).
Multiple critical points — sign-test each region
For a cubic f, the derivative f′ is quadratic with up to two roots. Factor f′, find its roots, then test the sign in each region between them. Mark the function as increasing or decreasing on each interval.
Polynomial degree of f
Degree of f′
Possible critical points
Test regions
quadratic
linear
1
2 (left of root, right of root)
cubic
quadratic
0, 1, or 2
up to 3
quartic
cubic
up to 3
up to 4
🧭 Recipe — find increasing/decreasing intervals
Differentiate to get f′(x).
For a single point: evaluate f′(x₁) and check sign.
For an interval: solve f′(x) > 0 (increasing) or f′(x) < 0 (decreasing).
For a polynomial f′: factor, find roots, test sign in each region between roots.
State the answer as inequalities or interval notation; combine with “or” for disjoint regions.
Worked examples
WE 1
Test increasing/decreasing at specific points
The function f(x) = x³ − 6x² + 5. Determine whether f is increasing or decreasing at: (a) x = 1; (b) x = 5.
Differentiatef′(x) = 3x² − 12x(a) Evaluate at x = 1f′(1) = 3 − 12 = −9−9 < 0 → f is decreasing at x = 1(b) Evaluate at x = 5f′(5) = 75 − 60 = 1515 > 0 → f is increasing at x = 5(a) decreasing at x = 1; (b) increasing at x = 5just check the SIGN of f′ — magnitude doesn’t matter for this question
WE 2
Find interval where increasing — quadratic
The function f(x) = 2x² − 8x + 3. Find the values of x for which f is increasing.
Differentiate and set f′(x) > 0f′(x) = 4x − 84x − 8 > 04x > 8x > 2f is increasing for x > 2parabola opens upward (a = 2 > 0) so f is increasing to the right of the vertex at x = 2
WE 3
Find interval where decreasing — quadratic
The function g(x) = 5 + 6x − x². Find the values of x for which g is decreasing.
Differentiate and set g′(x) < 0g′(x) = 6 − 2x6 − 2x < 0−2x < −6Divide by −2 — flip the inequalityx > 3g is decreasing for x > 3parabola opens downward (coeff of x² is −1) so g decreases to the right of the vertex at x = 3
WE 4
Cubic — both increasing AND decreasing intervals
The function h(x) = x³ − 3x² − 9x + 11. Find the intervals on which h is (a) increasing; (b) decreasing.
Differentiate and factorh′(x) = 3x² − 6x − 9 = 3(x² − 2x − 3) = 3(x − 3)(x + 1)Roots of h′: x = −1 and x = 3 — split domain into 3 regionsSign test in each regionx < −1: at x = −2: 3(4 + 2 − 3) = 9 → +−1 < x < 3: at x = 0: 3(0 − 0 − 3) = −9 → −x > 3: at x = 4: 3(16 − 8 − 3) = 15 → +Read off the intervals(a) increasing for x < −1 or x > 3; (b) decreasing for −1 < x < 3at x = −1 and x = 3, h is stationary (these are the local max and local min)
WE 5
Show a function is always increasing
Show that f(x) = x³ + 4x − 7 is increasing for all real x.
Differentiatef′(x) = 3x² + 4Argue f′(x) > 0 for ALL real x3x² ≥ 0 for all real x (a square is non-negative)→ 3x² + 4 ≥ 4 > 0 for all real xConclusionf′(x) > 0 for all real x → f is increasing on ℝf is increasing for all real x ∎when f′ has no real roots, it has constant sign — check the sign at any one point
WE 6
Real-world: profit function with bounded domain
A company’s profit (in thousand dollars) is modelled by P(t) = −t³ + 12t² − 27t + 50, where t is months since launch and 0 ≤ t ≤ 10.
(a) Find P′(t). (b) Find the time intervals on which the profit is increasing. (c) Find the time intervals on which the profit is decreasing.
(a) DifferentiateP′(t) = −3t² + 24t − 27 = −3(t² − 8t + 9)Find roots of P′(t) = 0 using the quadratic formulat² − 8t + 9 = 0t = (8 ± √(64 − 36))/2 = (8 ± √28)/2 = 4 ± √7→ t ≈ 1.354 and t ≈ 6.646Sign test (leading coeff −3, so parabola opens downward; P′ > 0 BETWEEN roots)at t = 0: P′ = −27 → − (decreasing)at t = 4: P′ = 21 → + (increasing)at t = 10: P′ = −87 → − (decreasing)Read off intervals (in domain [0, 10])(b) increasing for 4 − √7 < t < 4 + √7 (≈ 1.35 < t < 6.65) (c) decreasing for 0 ≤ t < 4 − √7 OR 4 + √7 < t ≤ 10profit climbs from month ~1.35 to month ~6.65, then falls — peaks near t ≈ 6.65
💡 Top tips
Sketch a sign chart for f′ — it makes intervals obvious for cubics or higher.
For a quadratic f′, the leading coefficient sign tells you whether f′ is positive between or outside its roots.
“Always increasing” proofs: if f′ is a sum of non-negatives plus a positive constant (like 3x² + 4), you’re done.
Boundary points: if the question gives a closed domain like [0, 10], include the endpoints in your final intervals.
Strict vs non-strict: at stationary points f′ = 0, so most exam answers use strict < and >.
⚠ Common mistakes
Sign confusion: f′(x) > 0 means INCREASING (NOT decreasing) — easy to flip under exam pressure.
Forgetting to test sign in each region between critical points — for cubics, you need at least 3 test values.
Stating x = 1 when an interval is asked for — “where increasing” needs a range, not a single value.
Forgetting to flip inequality when dividing by a negative — −2x < −6 gives x > 3, not x < 3.
Mixing up f and f′ — sign of f tells you the y-value; sign of f′ tells you whether y is rising or falling.
That closes the Differentiation introduction. Up next: Stationary Points and the second derivative test, where f′ = 0 (the boundaries between increasing and decreasing) get classified as local maxima, local minima, or points of inflection. After that comes the chain, product and quotient rules — the toolkit for differentiating composite, multiplied, and divided functions, including all the trig, exp and log standard derivatives.
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