Question

Use Newton's method to find the absolute maximum value of the function f(x) = 3x sin x, 0 ≤ x ≤ π correct to six decimal places.

Answer 1

f'(x) = 5sin(x) + 5x*cos(x) = 0.

Then, setting f'(x) = 0 gives:
5sin(x) + 5x*cos(x) = 0 ==> tan(x) + x = 0.

Then, use Newton's Method to solve tan(x) + x = 0 on 0 <= x <= π. This solves to get x ≈ 2.028758. Using the Second Derivative Test, you can show that this gives a maximum. Therefore, the required maximum value is:
f(2.028758) = 2(2.028758)sin(2.028758) ≈ 3.639411........google told me this