Using the bisection method, Newton’s method, and the secant method, find the largest

positive root correct to three decimal places of x3 − 5x + 3 = 0. (All roots are in
[−3,+3].)

To find the largest positive root of the equation x^3 - 5x + 3 = 0 using the bisection method, Newton's method, and the secant method, we can follow these steps:

1. Bisection Method:
a. Start by plotting the function f(x) = x^3 - 5x + 3.
b. Since we are looking for a positive root, we know it lies in the interval [0, 3]. Choose the initial interval a = 0 and b = 3.
c. Calculate the midpoint c = (a + b) / 2.
d. Evaluate f(c).
- If f(c) is close to zero (within a small tolerance), then c is our root.
- If f(c) has the same sign as f(a), update a = c.
- If f(c) has the same sign as f(b), update b = c.
e. Repeat steps c to d until you arrive at a root with the desired accuracy.

2. Newton's Method:
a. Start by differentiating the function f(x) = x^3 - 5x + 3 to find its derivative f'(x).
- f'(x) = 3x^2 - 5
b. Choose an initial guess x0 in the interval [0, 3].
c. Iterate using the formula: x1 = x0 - f(x0)/f'(x0).
d. Repeat step c until you arrive at a root with the desired accuracy.

3. Secant Method:
a. These are the steps to follow in the secant method:
- Start by choosing two initial guesses x0 and x1 in the interval [0, 3].
- Use the formula: x2 = x1 - [(x1 - x0) * f(x1)] / [f(x1) - f(x0)].
- Update x0 = x1 and x1 = x2.
- Repeat the above step until you arrive at a root with the desired accuracy.

By applying one of these methods with the desired accuracy, you can find the largest positive root of the equation x^3 - 5x + 3 = 0.

To find the largest positive root of the equation x3 - 5x + 3 = 0 using the bisection method, Newton's method, and the secant method, we will iterate through each method until we reach the desired accuracy.

Bisection Method:
1. Begin by identifying the interval [a, b] where the root exists. In this case, the interval is [-3, 3].
2. Calculate the midpoint c = (a + b) / 2.
3. Evaluate the function f(c) = c3 - 5c + 3.
4. If f(c) is close enough to 0 (within the desired accuracy), then we have found the root. Otherwise, continue to the next step.
5. Determine if the root is in the left or right subinterval. If f(a) * f(c) < 0, then the root is in the left subinterval. Otherwise, it is in the right subinterval.
6. Repeat steps 2-5 with the new interval until the root is found within the desired accuracy.

Newton's Method:
1. Start with an initial guess x0 within the interval [-3, 3].
2. Calculate the next guess x1 using the formula: x1 = x0 - f(x0) / f'(x0), where f'(x0) is the derivative of the function at x0.
3. Repeat step 2 until the guess converges to the root within the desired accuracy. The formula for the next guess becomes xn+1 = xn - f(xn) / f'(xn).

Secant Method:
1. Choose two initial guesses, x0 and x1, within the interval [-3, 3].
2. Calculate the next guess x2 using the formula: x2 = x1 - f(x1) * (x1 - x0) / (f(x1) - f(x0)).
3. Repeat step 2 with x2 as the new x1 and x1 as the new x0 until the guess converges to the root within the desired accuracy.

Now let's apply these methods step-by-step to find the largest positive root of the equation x3 - 5x + 3 = 0.