One real root of a polynomial P(x) = x³ - x - 1 lies in the interval [1,2] and bisection method is used to find its value, the minimum number of iterations required to achieve accuracy up to two decimal points is

Question

Question

Answer 1

Concept:

To achieve accuracy up to two decimal places means the error between the real root and numerically calculated root should be less than 0.01

⇒ ϵ < 0.01

f(x) = x³ – x – 1, 1 ≤ x ≤ 2

Accuracy (ε) = 0.01

Accuracy of bisection method is given by,

\(\varepsilon \ge \left( {\frac{{b - a}}{{{2^n}}}} \right)\)

\(\Rightarrow 0.01 \ge \left( {\frac{{2 - 1}}{{{2^n}}}} \right)\)

\(\Rightarrow 0.01 \ge \frac{1}{{{2^n}}}\)

⇒ 2n ≥ 100

Now 2⁶ = 64, 2⁷ = 128

∴ Minimum number of n to satisfy the above condition is, n = 7