Answer:
Here's the step-by-step solution using Newton's method:
Finding the critical points:
8cos(x) - 8xsin(x) = 0
cos(x) - xsin(x) = 0
Applying Newton's method:
Let's start with an initial guess x₀ = 1.
g(x) = cos(x) - xsin(x)
g'(x) = -sin(x) - sin(x) - xcos(x)
x₁ = x₀ - (g(x₀) / g'(x₀))
x₁ = 1 - (cos(1) - sin(1)) / (-sin(1) - sin(1) - cos(1))
x₁ ≈ 0.450183
Repeat the iteration until the desired level of accuracy is reached.
Iteration 2: x₂ ≈ 0.450184
Iteration 3: x₃ ≈ 0.450184
Evaluate f(x) at critical points and endpoints:
f(0) = 0
f(π) = -8π
Compare the values:
f(x) at the critical point x = 0 is 0.
f(x) at the critical point x ≈ 0.450184 is approximately 3.628269.
f(x) at the endpoint x = π is approximately -25.132741.
Therefore, the absolute maximum value of the function f(x) = 8xcos(x) on the interval 0 ≤ x ≤ π is approximately 3.628269, occurring at x ≈ 0.450184.