--- title: "Increasing and Decreasing Functions" description: "Use the sign of the derivative to say where a curve rises and where it falls, which turns a question about shape into an inequality in x." canonical: https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ source: https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Increasing and Decreasing Functions Use the sign of the derivative to say where a curve rises and where it falls, which turns a question about shape into an inequality in x. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ ## Ready when they can - Find the interval where y = x² - 6x is decreasing - Show that a given cubic is increasing for all x - Match the sign of f'(x) to the parts of a sketched curve ## Lesson: Increasing and Decreasing Functions Last stop you fed one x into dy/dx and got one gradient, then built a line from it. Now ask a bigger question. Walking the Ridgeway ramp from left to right, which stretches climb and which drop away? That is not about one point, it is about every point in a stretch at once. The tool is the same derivative, read for its sign rather than its size, and the answer will be a range of x values rather than a single number. The rule is short. If f'(x) is positive at every x in an interval, the curve rises across that whole interval, and the function is called increasing there. If f'(x) is negative throughout, it falls, and the function is decreasing there. So a question about the shape of a curve turns into a question about the sign of an expression, which is an inequality in x. You have solved those since the linear ones, and for a cubic the inequality that falls out is quadratic. *(drawing: The sign carries the shape. Solving for it is work you can already do.)* **Example.** Take y = x² - 6x, so dy/dx = 2x - 6. For decreasing, solve 2x - 6 < 0, which gives 2x < 6, so x < 3. For increasing, solve 2x - 6 > 0 instead, giving x > 3. Check it against the shape you know: this is an upward parabola, so it falls, then turns and climbs, and x = 3 is exactly where the two stretches meet. The derivative and the sketch are telling the same story in two languages. **Example.** Now y = x³ - 6x², so dy/dx = 3x² - 12x. Factorise before solving: 3x(x - 4), which is zero at x = 0 and x = 4. This is an upward parabola in x, so it sits above the axis outside those roots and below between them. So the curve is increasing for x < 0 and for x > 4, and decreasing for 0 < x < 4. Two separate increasing stretches from one cubic, which no linear derivative could ever produce. *(drawing: Factorise, find the roots, then use the parabola shape to pick the regions.)* Some curves never turn. Take y = x³ + 9x, so dy/dx = 3x² + 9. A square is never negative, so 3x² is at least 0 and the whole thing is at least 9. It is positive for every x, so this curve is increasing everywhere and has no falling stretch at all. One caution about the boundaries. Where f'(x) = 0 the curve is flat for an instant, and whether that spot is a peak, a trough or neither is a separate question for the next stop. **Recap.** The sign of the derivative says which way a curve is going. Where f'(x) > 0 across an interval the function is increasing; where f'(x) < 0 it is decreasing. So finding those stretches means solving an inequality: linear for a quadratic curve, quadratic for a cubic one. A derivative that is never negative, like 3x² + 9, means a curve that rises everywhere. What happens exactly at a boundary, where f'(x) = 0, is a question for the next stop. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp) - [Solving Quadratic Inequalities](https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8) ## Opens up - [Stationary Points and the Second Derivative](https://lightmysky.com/learn/mathematics/stationary-points-and-the-second-derivative-mt_VbGJEFFgfs)