Question

Solve f(x) = x³ + 12x² - 100x – 6 using false position with a = 5, b = 6, and e_s =0.5%. Show each step and create a table. Please be as detailed as you can.

Answer 1

At first, we need to verify that for given value of and :

This ensures that the function has opposite sign in the interval, this means it has crossed the y=0 line and contains a root within.

For your reference, the formula for calculating approximation of root at each iteration is:

As at root, the function should ideally take the value of '0', the tolerance of 0.5% is being taken as:

"the value of x at which the function satisfies:

First iteration:

a=5 => f(5.)=-81

b=6 =>    f(6)=42   

Putting these value in above formula gives:

This finally gives: x_n=5.658537 ,f(5.658537)=-6.444335, here: , hence we continue iteration

Now as f(x_n)f(b)<0 =>a=x_n =5.658537, for the next iteration

Second iteration:

a=5.658537    => f(5.658537)=-6.444335   

b=6 => f(6)=42   

x_n=5.703960 => f(5.703960)=-0.394841, here: , hence we continue iteration

Now as f(x_n)f(b)<0 =>a=x_n =5.703960 for the next iteration

Third iteration:

a=5.703960    => f(5.703960)=-0.394841   

b=6.000000    => f(6.000000)=42.000000

xn=5.706717   => f(5.706717)=-0.023782, here: , hence we continue iteration

Now as f(x_n)f(b)<0 =>a=x_n =5.706717 for the next iteration

Fourth iteration:

a=5.706717    => f(5.706717)=-0.023782   

b=6.000000    => f(6.000000)=42.000000  

xn=5.706883   => f(5.706883)=-0.001431, here: , hence we stop the iterations!

Now we have x_n =5.706883 as the root for our function!!!

Hope this helps!