Change of Sign Failure (AQA A-Level Mathematics): Quizzes

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Change of Sign Failure
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What is change of sign failure?

Method misses a root that exists

What happens at a repeated root like f(x)=(x−2)2f(x) = (x - 2)^2?

Function touches x-axis, no crossing

For f(x)=(x−2)2f(x) = (x - 2)^2, is there a sign change at x=2x = 2?

No, there is no sign change

Which methods can be more effective when change of sign fails?

Newton-Raphson or fixed-point iteration

If f(1)<0f(1) < 0 and f(2)<0f(2) < 0 with two roots in [1,2][1, 2], why does the method fail?

No change of sign occurs

For f(x) = rac{1}{x}, what occurs between f(−1)f(-1) and f(1)f(1)?

Sign change but no root

What condition must hold for the change of sign method to work?

Function must be continuous

Why does f(x) = rac{1}{x} show change of sign but no root?

Discontinuity at the y-axis

If f(1)>0f(1) > 0 and f(2)>0f(2) > 0 but a root exists in [1,2][1, 2], what is the issue?

No sign change to detect root

Which type of function typically causes change of sign failure?

Discontinuous or repeated root functions

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