10.1.2 Change of Sign Failure

Failures of the Change of Sign Method

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Change of sign failure occurs when the change of sign method does not detect a root, even though one exists. This typically happens in A Level Maths under the following conditions:

Repeated roots: The function touches the x-axis without crossing it, as in $f(x) = (x - 2)^2$ , where there's no sign change despite a root at $x = 2$ .
Discontinuous functions: If the function is not continuous, it may jump over the root, preventing a change of sign from being detected. In such cases, alternative numerical methods, like Newton-Raphson or fixed-point iteration, may be more effective for finding roots.

Multiple Roots Without a Sign Change:

Consider a function where $f(1)$ is negative and $f(2)$ is also negative.
Even though there are two roots within the interval, there is no change of sign.
Conclusion: The method fails because it doesn't detect the roots due to the absence of a sign change.

Discontinuities in the Function:

Example: $f(x) = \frac{1}{x}$
Here, $f(-1) < 0$ and $f(1) > 0$ , indicating a change of sign, but there is no root.
This occurs because the function has a discontinuity at the $y$ -axis.
Conclusion: The Change of Sign method only works when a function is continuous within a given interval. The method fails if there are discontinuities.

No Sign Change Despite a Root Existing:

Consider a function where $f(1) > 0$ and $f(2) > 2$ , indicating no change of sign.
However, a root exists within the interval $[1, 2]$ .
Conclusion: The method fails because it doesn't identify the root due to the lack of a sign change within the interval.

Change of Sign Failure Simplified Revision Notes for A-Level AQA Maths Pure

10.1.2 Change of Sign Failure

Failures of the Change of Sign Method

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