What must you state about f besides the sign change?

That f is continuous on the interval.

Which function types are always continuous?

Polynomials, e^x, \sin x, \cos x (and \ln x for x0).

Why does h(x)=\frac{1}{x-2} fail the test on [1,3]?

It is not continuous at x=2 (asymptote), so the sign change does not give a root.

Does f(a) and f(b) with the same sign prove there is no root?

No. There could be an even number of roots (or a repeated root) in the interval.

Show that x^3+x-5=0 has a root in [1,2]: what values?

f(1)=-3 and f(2)=5: a change of sign.

How do you show a root is 3.5 to 1 d.p.?

Show a sign change between 3.45 and 3.55.

What does a repeated root, such as (x-2)^2=0, do to the sign?

It touches the axis without a sign change.

What is a good first step to find an interval for a root?

Tabulate f(x) at integer values and look for a sign change.

Location of rootsEdexcel International A Level Maths: Flashcards

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State the change-of-sign test for a root.

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State the change-of-sign test for a root.: If $f$ is continuous on $[a,b]$ and $f(a)$ , $f(b)$ have opposite signs, then $f(x)=0$ has a root in the interval.
What must you state about $f$ besides the sign change?: That $f$ is continuous on the interval.
Which function types are always continuous?: Polynomials, $e^x$ , $\sin x$ , $\cos x$ (and $\ln x$ for $x>0$ ).
Why does $h(x)=\frac{1}{x-2}$ fail the test on $[1,3]$ ?: It is not continuous at $x=2$ (asymptote), so the sign change does not give a root.
Does $f(a)$ and $f(b)$ with the same sign prove there is no root?: No. There could be an even number of roots (or a repeated root) in the interval.
Show that $x^3+x-5=0$ has a root in $[1,2]$ : what values?: $f(1)=-3$ and $f(2)=5$ : a change of sign.
How do you show a root is $3.5$ to 1 d.p.?: Show a sign change between $3.45$ and $3.55$ .
What unit must a calculator use for trigonometric roots?: Radians.
A root is found in $[1.51,1.52]$ . What is it to 1 d.p.?: $1.5$
What does a repeated root, such as $(x-2)^2=0$ , do to the sign?: It touches the axis without a sign change.
How many distinct roots can a cubic have at most?: Three.
What is a good first step to find an interval for a root?: Tabulate $f(x)$ at integer values and look for a sign change.

Exam questions on Location of roots

The function $f$ is defined by $f(x)=x^3+x-5$ for $x\in\mathbb{R}$ . The equation $f(x)=0$ has exactly one real root $\alpha$ .
Given that $f(1.6)=0.696$ , explain how the values of $f(1.5)$ and $f(1.6)$ show that $1.5<\alpha<1.6$ .2 marks
The functions $h$ and $p$ are defined by $h(x)=\frac{1}{x-2}$ for $x\neq2$ , and $p(x)=x^2-2$ for $x\in\mathbb{R}$ .
A student notes that $h(1)<0$ and $h(3)>0$ , and concludes that $h(x)=0$ has a root in $[1,3]$ . Explain why this conclusion is wrong.2 marks
The function $f$ is defined by $f(x)=e^x-3x$ for $x\in\mathbb{R}$ . The equation $f(x)=0$ has two real roots, $\alpha$ and $\beta$ , with $\alpha<\beta$ .
Show that $\alpha$ lies in the interval $[0.5,0.7]$ .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).