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MathidiaMathematics Academy
Numerical methods
Lesson 36 of 75
Introduction

Change of sign

Objective. Prove a root lies in a stated interval.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

The change of sign method proves that a root of f(x) = 0 lies within a given interval by showing f(x) changes from positive to negative (or vice versa) across it, implying a root exists by the intermediate value theorem for a continuous function. For f(x) = x³ − x − 1, f(1) = −1 and f(2) = 5, and since f is continuous with a sign change, a root lies between x = 1 and x = 2. Always state the function is continuous on the interval before concluding a root exists.

Worked example

Show by a change of sign that f(x) = x³ − 4x − 2 has a root between x = 2 and x = 3.

4 … 2
Illustrated card showing 4 … 2
  1. 1Evaluate f(2) = 8 − 8 − 2 = −2
  2. 2Evaluate f(3) = 27 − 12 − 2 = 13
  3. 3Note f(2) < 0 and f(3) > 0, a change of sign
  4. 4Since f is continuous, a root lies between x = 2 and x = 3

Answer: Root confirmed between x = 2 and x = 3 since f(2) = −2 and f(3) = 13

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Common mistake. Students often conclude a root exists without checking continuity, or without evaluating f at both endpoints exactly. A sign change alone, with continuity stated, is required; a single value close to zero is not sufficient evidence.

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