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10.1PureCore

Locating Roots

Many equations cannot be solved exactly. The first step in a numerical approach is to locate a root: show that a continuous function changes sign across an interval, so a root must lie inside it.

24 min Numerical Methods
Edexcel A level Maths: 10.1 Locating RootsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Use the sign-change rule
  • Show a root lies in an interval
  • Understand when sign change fails
  • Connect roots to f(x) = 0
1

The sign-change rule

If ff is continuous on [a,b][a, b] and f(a)f(a) and f(b)f(b) have opposite signs\textbf{opposite signs}, then there is at least one root of f(x)=0f(x) = 0 in (a,b)(a, b).

f(a) f(b)<0⇒root in (a,b)f(a)\,f(b) < 0 \Rightarrow \text{root in } (a,b)
Continuous f with a sign change.
Example 1Show f(x)=x3−x−1f(x) = x^3 - x - 1 has a root between 1 and 2.
1f(1)=−1<0f(1) = -1 < 0, f(2)=5>0f(2) = 5 > 0.
2Sign change and f continuous, so a root lies in (1,2)(1,2).
Answer
Root in (1, 2).

Tip — Always state that f is continuous — the rule needs it.

2

When it fails

A sign change guarantees a root, but no sign change does NOT guarantee none (there may be an even number of roots). Also, if ff is discontinuous (e.g. an asymptote) a sign change may not indicate a root.

Formula recap

f(a) f(b)<0⇒root in (a,b)f(a)\,f(b) < 0 \Rightarrow \text{root in } (a,b)
Sign-change rule (f continuous).

Common mistakes to avoid

Forgetting to check continuity.
The sign-change rule requires f continuous on the interval.
Concluding “no root” from no sign change.
There could be an even number of roots; no sign change is inconclusive.

Key takeaways

  • Sign change of continuous f over [a,b] ⇒ root in (a,b).
  • State continuity explicitly.
  • No sign change is not proof of no root.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A root lies in (a,b) if f is continuous and:

    • f(a)f(b)<0f(a)f(b) < 0
    • f(a)f(b)>0f(a)f(b) > 0
    • f(a)=f(b)f(a) = f(b)
    • f(a)>0f(a) > 0
    Show answer and explanation

    Answer: f(a)f(b)<0f(a)f(b) < 0

    Opposite signs.

  2. The sign-change rule requires f to be:

    • continuous
    • linear
    • increasing
    • positive
    Show answer and explanation

    Answer: continuous

    Else it can fail.

  3. For f(x)=x3−x−1f(x) = x^3 - x - 1, f(1)f(1) and f(2)f(2) are:

    • −1-1 and 55
    • 11 and 55
    • −1-1 and −5-5
    • 00 and 55
    Show answer and explanation

    Answer: −1-1 and 55

    Compute each.

  4. No sign change across [a,b] means:

    • inconclusive
    • definitely no root
    • definitely a root
    • two roots
    Show answer and explanation

    Answer: inconclusive

    Could be an even number.

  5. A root of f corresponds to:

    • f(x)=0f(x) = 0
    • f′(x)=0f'(x) = 0
    • f(x)=1f(x) = 1
    • a maximum
    Show answer and explanation

    Answer: f(x)=0f(x) = 0

    Definition.

Next step

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