A function is increasing where its gradient is positive, f′(x) > 0, and decreasing where its gradient is negative, f′(x) < 0. To find the intervals, differentiate, factorise f′(x) and solve the inequality. For a cubic, f′(x) is a quadratic, so the decreasing interval usually lies between its roots and the increasing intervals lie outside them.
Increasing and decreasing
- Increasing: f′(x) > 0
- Decreasing: f′(x) < 0
- Stationary: f′(x) = 0, where the curve switches between the two
A U-shaped quadratic is negative between its roots and positive outside them.
How does the gradient show increasing and decreasing?
The gradient f′(x) is positive where the curve rises and negative where it falls. So to find where a function is increasing, differentiate and solve f′(x) > 0; to find where it is decreasing, solve f′(x) < 0. The answers are inequalities in x.
Decreasing intervals of a cubic
For a cubic, f′(x) is a quadratic. Factorise it and find its roots. If the x² coefficient is positive, the quadratic dips below zero between its roots, so the cubic is decreasing there. For example, f′(x) = 3x(x − 2) is negative for 0 < x < 2.
Increasing intervals of a cubic
The same factorised f′(x) is positive outside its roots, so the cubic is increasing on two separate intervals. Write the answer as a pair joined by “or”, such as x < −1 or x > 1, not as a single range. Some questions accept the endpoints included, using ≤ and ≥; follow the wording of the question.
Showing a function is increasing for all x
A function is increasing for all x if f′(x) > 0 for every x. Show this by writing f′(x) as a square plus a positive constant, or by completing the square, or by showing the quadratic f′(x) has a negative discriminant and a positive leading coefficient. Since x² ≥ 0, an expression like 3x² + 3 is always at least 3.
Linking to stationary points
A curve switches between increasing and decreasing at its stationary points, where f′(x) = 0. So once f′(x) is factorised, its roots give the turning points, and the sign of f′(x) on each side gives the intervals.
Worked examples
Example 1. Find the values of x for which y = x² − 6x is increasing.
- f′(x) = 2x − 6.
- Increasing: 2x − 6 > 0, so x > 3.
Answer: x > 3.
Marks: M1 for 2x − 6 > 0, A1 for x > 3.
Example 2. Find the interval on which y = x³ − 3x² is decreasing.
- f′(x) = 3x² − 6x = 3x(x − 2).
- Decreasing: 3x(x − 2) < 0, which is between the roots 0 and 2.
Answer: 0 < x < 2.
Marks: M1 for the inequality, A1 for 0 < x < 2.
Example 3. Find the values of x for which y = x³ − 3x is increasing.
- f′(x) = 3x² − 3 = 3(x − 1)(x + 1).
- Increasing: 3(x − 1)(x + 1) > 0, which is outside the roots.
Answer: x < −1 or x > 1.
Marks: M1 for the inequality, A1 for both intervals.
Example 4. Show that y = x³ + 3x is increasing for all x.
- f′(x) = 3x² + 3.
- x² ≥ 0 for all x, so 3x² + 3 ≥ 3 > 0.
Answer: f′(x) > 0 for all x, so the function is always increasing.
Marks: M1 for f′(x), A1 for the argument.
Example 5. A function has f′(x) = (x − 1)(x − 5). State where f is increasing and where it is decreasing.
- The roots are x = 1 and x = 5: these are the turning points.
- f′(x) is positive outside the roots and negative between them.
Answer: Increasing for x < 1 or x > 5; decreasing for 1 < x < 5.
Marks: M1 for the roots, A1 for both intervals.
Common mistakes
- Giving a single range for two intervals. Outside the roots means x < a or x > b, two separate intervals.
- Swapping between and outside. For a U-shaped f′(x), negative values are between the roots.
- Not factorising f′(x). Factorise first, so the roots and signs are clear.
- Writing the answer in terms of y. Increasing and decreasing intervals are ranges of x.
- Not justifying “for all x”. State why f′(x) is always positive, for example because x² ≥ 0.
How increasing and decreasing functions are examined
This topic sits on the Pure 2 paper (WMA12) of Edexcel International A-Level and in Year 1 of Edexcel A-Level (9MA0). Typical question parts:
- find dy/dx and show it factorises in a given way
- find the set of values of x for which a curve is increasing or decreasing
- show a function is increasing (or decreasing) for all x
- connect the intervals to the turning points
Multi-part questions with mark schemes are in the Pure 2 differentiation exam questions. Try the Pure 2 differentiation worksheet too, or the differentiation revision engine.
You need quadratic inequalities here, and the next step is second derivatives and stationary points.
Frequently asked questions
How do you know if a function is increasing?
Differentiate. Where f′(x) > 0 the function is increasing, and where f′(x) < 0 it is decreasing.
How do you show a function is increasing for all x?
Show that f′(x) is positive for every x, for example by writing it as a square plus a positive number.
Where does a function change from increasing to decreasing?
At a stationary point, where f′(x) = 0 and the sign of f′(x) changes.