Final answer:
To find the absolute maximum of the function f(x) = 3x sin x on the interval [0, π], we calculate the derivative and set it equal to zero, find critical points, and evaluate the function at these points and at the interval endpoints. Newton's method is not directly used here, as we are finding the derivative to locate maximum values.
Step-by-step explanation:
To find the absolute maximum value of the function f(x) = 3x sin x on the interval [0, π], we first need to determine the critical points of the function within this interval where the function's first derivative is zero or undefined.
The first derivative of f(x) is:
f'(x) = 3 sin x + 3x cos x
Setting the derivative equal to zero to find the critical points:
0 = 3sin x + 3x cos x
The values of x that satisfy the above equation can potentially be points where the maximum value occurs, but we also consider the endpoints x = 0 and x = π. Evaluating the function at these points as well as at any critical points between them will determine the absolute maximum on the given interval.
Note: Newton's method is typically used to approximate the roots of a function, not to directly find maximum values. In this case, we're using calculus to find the critical points directly. To get the answer correct to six decimal places, one would have to use numerical methods or a high-precision calculator after finding the critical points, as analytic solutions may not yield such precision.