To find the smallest positive root of the equation $f(x) = x^3 - 9x + 1 = 0$ using the bisection method, follow these steps:

Step 1: Identify the interval containing the root

We need to find an interval $[a, b]$ such that $f(a) \cdot f(b) < 0$, indicating a root exists in $[a, b]$ (by the Intermediate Value Theorem).

Step 2: Initial check of values

Evaluate $f(x)$ at small positive integers to locate the root's approximate range:

• $f(0) = 0^3 - 9 \cdot 0 + 1 = 1$
• $f(1) = 1^3 - 9 \cdot 1 + 1 = -7$
• $f(2) = 2^3 - 9 \cdot 2 + 1 = -9$
• $f(3) = 3^3 - 9 \cdot 3 + 1 = 1$

The sign change between $x = 2$ and $x = 3$ confirms that the root lies in $[2, 3]$.

Step 3: Apply the Bisection Method

1. Compute the midpoint $c = \frac{a + b}{2}$ and evaluate $f(c)$.
2. Replace the interval's endpoint based on the sign of $f(c)$:
   • If $f(a) \cdot f(c) < 0$, set $b = c$.
   • If $f(c) \cdot f(b) < 0$, set $a = c$.
3. Repeat until the length of the interval is smaller than the desired accuracy (0.01).

Step 4: Iterative Calculations

Iteration 1