Science:Math Exam Resources/Courses/MATH103/April 2017/Question 07 (b)/Solution 1

When $|f^{\prime }(x_0)|=1$, 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 $|f^{\prime }(x_0)|<1$, and therefore that the the fixed point is stable; a line that is steeper than $y=x$ satisfies $|f^{\prime }(x_0)|>1$ The coordinates for 4 fixed points is $(-5,-5),(0,0),(5,5),(10,10)$.

Answer:

point	stability	reason
(i) $(-5,5)$	unstable	$f^{\prime }(-5)>1$
(ii) $(0,0)$	stable	$f^{\prime }(0)=0<1$
(iii) $-(5,5)$	unstable	$f^{\prime }(5)>1$
(iv) $(10,10)$	stable	$f^{\prime }(10)=0<1$