Increasing & Decreasing Functions (AQA GCSE Further Maths): Revision Note

• 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

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

The function is decreasing when its gradient $\left(\frac{dy}{dx}\right)$ is less than 0.

Find the derivative of the function by differentiating.

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

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

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

Factorise.

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

The solutions to $\frac{dy}{dx}=0$ are $x=0$ and  $x=1$. Find the correct way around for the inequalities by considering the graph of $\frac{dy}{dx}$. 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.

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

$\mathbf{0}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{1}$