What does it mean for f to be decreasing on an interval?

f(x) gets smaller as x gets larger; the graph falls from left to right.

What does f'(x)0 tell you?

f is increasing at x (tangent slopes upwards).

What does f'(x)<0 tell you?

f is decreasing at x (tangent slopes downwards).

What does f'(x)=0 tell you?

The tangent is horizontal; the graph is momentarily flat. These x-values are the interval boundaries.

Four steps to find where f is increasing or decreasing?

Find f'(x); solve f'(x)=0; check the sign of f' in each region; write the intervals in x.

How do you check the sign of f' in a region?

Substitute a test value from inside the region into f'(x), or sketch the graph of f'.

f'(x)=(x-1)(x-5). Where is f decreasing?

1<x<5 (between the roots, as f' is an upward-opening quadratic).

On the graph of f', where is f increasing?

Where the graph of f' is above the x-axis.

Are intervals of increase and decrease written in x or y?

In x, using inequalities such as -1<x<3.

Can a function be both increasing and decreasing?

Yes, on different intervals, so list every interval.

5.2 Increasing and decreasing functionsIB Maths: Applications and Interpretation SL: Flashcards

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What does it mean for $f$ to be increasing on an interval?

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What does it mean for $f$ to be increasing on an interval?: $f(x)$ gets larger as $x$ gets larger; the graph rises from left to right.
What does it mean for $f$ to be decreasing on an interval?: $f(x)$ gets smaller as $x$ gets larger; the graph falls from left to right.
What does $f'(x)>0$ tell you?: $f$ is increasing at $x$ (tangent slopes upwards).
What does $f'(x)<0$ tell you?: $f$ is decreasing at $x$ (tangent slopes downwards).
What does $f'(x)=0$ tell you?: The tangent is horizontal; the graph is momentarily flat. These $x$ -values are the interval boundaries.
Four steps to find where $f$ is increasing or decreasing?: Find $f'(x)$ ; solve $f'(x)=0$ ; check the sign of $f'$ in each region; write the intervals in $x$ .
How do you check the sign of $f'$ in a region?: Substitute a test value from inside the region into $f'(x)$ , or sketch the graph of $f'$ .
$f'(x)=(x-1)(x-5)$ . Where is $f$ decreasing?: $1<x<5$ (between the roots, as $f'$ is an upward-opening quadratic).
On the graph of $f'$ , where is $f$ increasing?: Where the graph of $f'$ is above the $x$ -axis.
Are intervals of increase and decrease written in $x$ or $y$ ?: In $x$ , using inequalities such as $-1<x<3$ .
If $g$ is decreasing on an interval and $a<b$ lie in it, compare $g(a)$ and $g(b)$ .: $g(a)>g(b)$ .
Can a function be both increasing and decreasing?: Yes, on different intervals, so list every interval.

Exam questions on 5.2 Increasing and decreasing functions

A function $f$ has derivative $f'(x)=x^2+x-6$ .
Find the set of values of $x$ for which $f$ is increasing.2 marks
A continuous function $g$ is defined for $-4\le x\le 6$ . Its derivative satisfies $g'(x)>0$ for $-4<x<-1$ , $g'(x)=0$ at $x=-1$ and at $x=3$ , $g'(x)<0$ for $-1<x<3$ , and $g'(x)>0$ for $3<x<6$ .
Given that $g(-1)=7$ , explain why $g(2)<7$ .2 marks
The height, $h$ metres, of a drone $t$ seconds after take-off is modelled by $h(t)=t^3-12t^2+36t+5$ , for $0\le t\le 8$ .
Find $h'(t)$ . Hence use your GDC to find the values of $t$ for which $h'(t)=0$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).