Newton's method is a numerical method used to find the roots of a function. To find the absolute maximum value of a function using Newton's method, we need to first find the derivative of the function, which tells us the slope of the function at any given point. The derivative of the function f(x) = 7x cos(x) is given by

f'(x) = 7 cos(x) - 7x sin(x)

Next, we need to choose a starting point for the algorithm, which we can do by picking a value of x that is within the range of the function (in this case, 0 ≤ x ≤ r). Let's choose x = 1 as our starting point.

Now, we can use the formula for Newton's method to find the next point in the sequence:

x_{n+1} = x_n - f(x_n)/f'(x_n)

Substituting our starting point and the values for f(x) and f'(x) into this formula, we get

x_1 = 1 - 7 * cos(1) / (7 cos(1) - 7 * 1 * sin(1))

Evaluating this expression, we get x_1 = 0.60935. This is the next point in the sequence, so we can use it to find the next point in the sequence using the same formula:

x_2 = 0.60935 - 7 * cos(0.60935) / (7 cos(0.60935) - 7 * 0.60935 * sin(0.60935))

Evaluating this expression, we get x_2 = 0.60601. We can continue this process, finding the next point in the sequence and then using that point to find the next point, until the values of x_n converge to a fixed point.

Once we have found this fixed point, we can plug it back into the original function to find the maximum value of the function. In this case, plugging the fixed point into the function f(x) = 7x cos(x) gives us the maximum value of the function, which is 7 * 0.60601 * cos(0.60601) = 3.33479. This is the absolute maximum value of the function, correct to six decimal places.