Find the fixed points of f(x) = 2x - x3 and determine their stability.

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.

the answer is; biodiversity resulting from few ancestors

adaptive radiation occurs when a population is subjected to a large natural environment with spatial ecological variations. the parts of the population in the different ecologies begin to adapt to their local environments. due to non-random mating across the large populations, this causes the different populations from the large population to speciate within their local environments. they, therefore, diverge to different species sharing a common ancestor.  

47 - 90 = 43

43 - 180 = x

x = 137