When
, the slope of that point should be parallel to
or
. This means that a line that is flatter than
satisfies
, and therefore that the the fixed point is stable; a line that is steeper than
satisfies
The coordinates for 4 fixed points is
.
Answer:
point
|
stability
|
reason
|
(i) ![{\displaystyle (-5,5)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d33cf5b87cfca7e6cbd4258fb4db96f39dfd1974) |
unstable |
![{\displaystyle f'(-5)>1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b9907beb46a9728c0d818d53396983c61489703b) |
|
(ii) ![{\displaystyle (0,0)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a) |
stable |
![{\displaystyle f'(0)=0<1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6318c1de6c7c685d0e16216f97529bc132a424e6) |
|
(iii) ![{\displaystyle -(5,5)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/e0f19719856ea6d8c497afaac0774eb1c06335e7) |
unstable |
![{\displaystyle f'(5)>1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/bc061b421466f17644d1c0545535dac0991470dc) |
|
(iv) ![{\displaystyle (10,10)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/24d992eb9ba6d7bbb8cd25cc966ce060fe220117) |
stable |
![{\displaystyle f'(10)=0<1}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b5c97e993876b017f3f4aca9335fcbd56ab77fde) |
|