Increasing & Decreasing Functions

What are increasing and decreasing functions?

A function is increasing when $\frac{d y}{d x} > 0$ (the gradient is positive)
- This means graph of a function goes up as $x$ increases
A function is decreasing when $\frac{d y}{d x} < 0$ (the gradient is negative)
- This means graph of a function goes down as $x$ increases

Incr Decr Illustr 1, A Level & AS Maths: Pure revision notes

To identify the intervals (the range of $x$ values) for which a curve is increasing or decreasing you need to:

1. Find the derivative $\frac{d y}{d x}$
2. Solve the inequalities $\frac{d y}{d x} > 0$ (for increasing intervals) or $\frac{d y}{d x} < 0$ (for decreasing intervals)

Exam Tip

In an exam, if you need to show a function is increasing or decreasing you can use either strict (<, >) or non-strict (≤, ≥) inequalities
- You will get the marks either way in this course

Worked example

For what values of $x$ is $y = 2 x^{3} - 3 x^{2} + 5$ a decreasing function?

The function is decreasing when its gradient $(\frac{d y}{d x})$ is less than 0.

Find the derivative of the function by differentiating.

$\begin{array}{rcl} \frac{d y}{d x} & = & 6 x^{2} - 6 x \end{array}$

Solve the inequality $\frac{d y}{d x} < 0$ to find the set of values where the gradient is negative.

$\begin{array}{rcl} 6 x^{2} - 6 x & < & 0 \end{array}$

Factorise.

$\begin{array}{rcl} 6 x (x - 1) & < & 0 \end{array}$

The solutions to $\frac{d y}{d x} = 0$ are $x = 0$ and $x = 1$ . Find the correct way around for the inequalities by considering the graph of $\frac{d y}{d x}$ . The graph is a positive quadratic, so the function is negative between the values of 0 and 1 (where it is below the $x$ -axis).

Considering a sketch of the graph of the gradient function may help you see this.

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You can check your answers by considering a sketch of the original function, it should be decreasing at the point where $0 < x < 1 .$

aqa-fm-increasing-and-decreasing-functions-rn-diagram-we-2

$0 < x < 1$

Increasing & Decreasing Functions (AQA GCSE Further Maths)

Revision Note