Science:Math Exam Resources/Courses/MATH103/April 2017/Question 07 (b)

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Question 07 (b)
The figure below shows the graph $y=f(x)$ and the diagonal $y=x$ (dashed line). Use this plot to answer the following questions: (b) Determine the stability of each fixed point. (Explain your reasoning.)

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Hint
A fixed point $x_{0}$ of $f$ is stable if and only if $\|f'(x_{0})\|<1$ .

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Solution

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When $|f'(x_{0})|=1$ , the slope of that point should be parallel to $y=x$ or $y=-x$ . This means that a line that is flatter than $y=x$ satisfies $|f'(x_{0})|<1$ , and therefore that the the fixed point is stable; a line that is steeper than $y=x$ satisfies $|f'(x_{0})|>1$ The coordinates for 4 fixed points is $(-5,-5),(0,0),(5,5),(10,10)$ .