5.2 Increasing and decreasing functionsIB Maths: Applications and Interpretation HL: Revision notes

A function $f$ is increasing on an interval if $f(x)$ gets larger as $x$ gets larger: its graph rises from left to right. It is decreasing on an interval if $f(x)$ gets smaller as $x$ gets larger: its graph falls from left to right. Intervals are always written in terms of $x$, for example 'increasing for $x>2$' or 'decreasing for $-1<x<3$'. The same function can be increasing on some intervals and decreasing on others, so a full answer lists all of them.

The derivative $f'(x)$ is the gradient of the tangent to the curve at $x$, so its sign tells you the direction of the graph:

• $f'(x)>0$: $f$ is increasing (tangent slopes upwards).
• $f'(x)<0$: $f$ is decreasing (tangent slopes downwards).
• $f'(x)=0$: the tangent is horizontal, so the graph is momentarily flat at that point. Values of $x$ where this happens are often the boundaries between intervals.

To find where $f$ is increasing or decreasing:

1. Find $f'(x)$.
2. Solve $f'(x)=0$ (use your GDC for anything awkward). These are the boundaries.
3. Find the sign of $f'(x)$ in each region, using a test value or a sketch of $f'$.
4. Write the intervals using inequalities in $x$. Example: $f(x)=x^3-3x^2-9x+2$. Then $f'(x)=3x^2-6x-9=3(x-3)(x+1)$, which is $0$ at $x=-1$ and $x=3$. Since $f'$ is an upward-opening quadratic, $f'>0$ outside the roots and $f'<0$ between them. So $f$ is increasing for $x<-1$ and $x>3$, and decreasing for $-1<x<3$. Check: $f'(0)=-9<0$.

From a graph of $f$: where the curve rises, $f'>0$; where it falls, $f'<0$; where it is flat at the top of a hill or bottom of a valley, $f'=0$. From a graph of $f'$ (the gradient graph): $f$ is increasing where the graph of $f'$ is above the $x$-axis, and decreasing where it is below. The points where the graph of $f'$ crosses the $x$-axis are the boundaries. Do not mix the two graphs up: a point high up on the graph of $f'$ means a steeply rising $f$, not a high value of $f$.

In modelling questions the independent variable is often time $t$. If $R'(t)>0$ for $3<t<7$, then $R$ is increasing between years $3$ and $7$: the quantity is growing. If $R'(t)<0$ the quantity is shrinking. To compare values, use the intervals: if $g$ is decreasing on an interval, then for two $x$-values $a<b$ in it, $g(a)>g(b)$. Whether an end-point such as $t=3$ is written with $<$ or $\le$ makes no difference to the marks, but the interval must be correct. Always answer in the context of the question and use units where given.