Find `root(3)(48)` using False Position method (regula falsi method)

Solution:
Let `x=48^(1/3)`

`:.x^3=48`

`:.x^3-48=0`

i.e. `f(x)=x^3-48`

Here

`x`	0	1	2	3	4
`f(x)`	-48	-47	-40	-21	16

`1^(st)` iteration :

Here `f(3) = -21 < 0` and `f(4) = 16 > 0`

`:.` Now, Root lies between `x_0 = 3` and `x_1 = 4`

`x_2 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`

`x_2=3 - (-21) * (4 - 3)/(16 - (-21))`

`x_2=3.5676`

`f(x_2)=f(3.5676)=3.5676^3-48=-2.5936 < 0`


`2^(nd)` iteration :

Here `f(3.5676) = -2.5936 < 0` and `f(4) = 16 > 0`

`:.` Now, Root lies between `x_0 = 3.5676` and `x_1 = 4`

`x_3 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`

`x_3=3.5676 - (-2.5936) * (4 - 3.5676)/(16 - (-2.5936))`

`x_3=3.6279`

`f(x_3)=f(3.6279)=3.6279^3-48=-0.2513 < 0`


`3^(rd)` iteration :

Here `f(3.6279) = -0.2513 < 0` and `f(4) = 16 > 0`

`:.` Now, Root lies between `x_0 = 3.6279` and `x_1 = 4`

`x_4 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`

`x_4=3.6279 - (-0.2513) * (4 - 3.6279)/(16 - (-0.2513))`

`x_4=3.6336`

`f(x_4)=f(3.6336)=3.6336^3-48=-0.0237 < 0`


`4^(th)` iteration :

Here `f(3.6336) = -0.0237 < 0` and `f(4) = 16 > 0`

`:.` Now, Root lies between `x_0 = 3.6336` and `x_1 = 4`

`x_5 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`

`x_5=3.6336 - (-0.0237) * (4 - 3.6336)/(16 - (-0.0237))`

`x_5=3.6342`

`f(x_5)=f(3.6342)=3.6342^3-48=-0.0022 < 0`


`5^(th)` iteration :

Here `f(3.6342) = -0.0022 < 0` and `f(4) = 16 > 0`

`:.` Now, Root lies between `x_0 = 3.6342` and `x_1 = 4`

`x_6 = x_0 - f(x_0) * (x_1 - x_0)/(f(x_1) - f(x_0))`

`x_6=3.6342 - (-0.0022) * (4 - 3.6342)/(16 - (-0.0022))`

`x_6=3.6342`

`f(x_6)=f(3.6342)=3.6342^3-48=-0.0002 < 0`


Approximate root of the equation `x^3-48=0` using False Position method is `3.6342` (After 5 iterations)

`n`	`x_0`	`f(x_0)`	`x_1`	`f(x_1)`	`x_2`	`f(x_2)`	Update
1	3	-21	4	16	3.5676	-2.5936	`x_0 = x_2`
2	3.5676	-2.5936	4	16	3.6279	-0.2513	`x_0 = x_2`
3	3.6279	-0.2513	4	16	3.6336	-0.0237	`x_0 = x_2`
4	3.6336	-0.0237	4	16	3.6342	-0.0022	`x_0 = x_2`
5	3.6342	-0.0022	4	16	3.6342	-0.0002	`x_0 = x_2`

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