Find the x-coordinates where f '(x) = 0 for f(x) = 2x + sin(2x) in the interval [0, 2π].
MY ANSWERS:
x = π/2
x = 3π/2
looks good to me
To find the x-coordinates where f'(x) = 0 for the function f(x) = 2x + sin(2x) in the interval [0, 2π], we need to find the critical points of the function.
Step 1: Calculate the derivative of f(x) with respect to x.
f'(x) = 2 + 2cos(2x)
Step 2: Set f'(x) = 0 and solve for x.
2 + 2cos(2x) = 0
cos(2x) = -1/2
Step 3: Find the angles x in the interval [0, 2π] where cos(2x) = -1/2.
To find the angles x, we need to consider the properties of the cosine function. Specifically, cos(θ) = -1/2 has solutions at θ = π/3 and θ = 5π/3.
Dividing these solutions by 2, we get:
x = π/6 and x = 5π/6
However, we need to make sure that these angles are within the interval [0, 2π].
For x = π/6, since π/6 < 2π, this solution is valid.
For x = 5π/6, since 5π/6 > 2π, this solution is not valid.
So the only x-coordinate within the interval [0, 2π] where f'(x) = 0 is x = π/6.