Answer with explanation:
⇒ A point k is said to be fixed point of function, f(x) if
f(k)=k.
The given function is
f(x)=2 x - x³
To determine the fixed point
f(k)=2 k - k³=k
→2 k -k -k³=0
→k -k³=0
→k×(1-k²)=0
→k(k+1)(k-1)=0
→k=0 ∧ k+1=0∧k-1=0
→k=0∧ k= -1 ∧ k=1
So, the three fixed points are=0,1 and -1.
To Check Stability of fixed point
1.⇒ f'(x)=2-3 x²
|f'(0)|=|2×0-0³|=0
⇒x=0, is Superstable point.
2.⇒|f'(-1)|=2 -3×(-1)²
=2 -3
= -1
|f'(-1)| <1
⇒x= -1, is stable point.
3.⇒|f'(1)|=2 -3×(1)²
=2 -3
= -1
|f'(1)| <1
⇒x= 1, is also a stable point.
⇒⇒There are two points of Stability,which are, x=1 and , x=-1.
