The interval that contains the root for the function f(x) = sin(x) is 3.
To determine which of the four intervals will contain the root for the function f(x) = sin(x) between x = 2 and x = 4 using four intervals, we can analyze the behavior of the sine function within each interval.
Interval 1: [2, 2.5]
Within this interval, the sine function is decreasing from sin(2) ≈ 0.906 to sin(2.5) ≈ 0.598.
Since sin(2.5) > 0, the root cannot lie in this interval.
Interval 2: [2.5, 3]
Within this interval, the sine function is changing from decreasing to increasing.
Specifically, it reaches a minimum value at x ≈ 2.708, where sin(2.708) ≈ -0.087.
This indicates that the root could potentially lie within this interval.
Interval 3: [3, 3.5]
Within this interval, the sine function is increasing from sin(3) ≈ -0.841 to sin(3.5) ≈ 0.355.
Since sin(3) < 0 and sin(3.5) > 0, the root must lie within this interval.
Interval 4: [3.5, 4]
Within this interval, the sine function is increasing from sin(3.5) ≈ 0.355 to sin(4) ≈ 0.657.
Since the root has already been found in the previous interval, it cannot lie in this interval.
Therefore, the root for the function f(x) = sin(x) between x = 2 and x = 4 will lie in Interval 3: [3, 3.5].