Question

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Find the x-coordinates where f '(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, π]. (10 points)

Answer 1

To find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, π], differentiate f(x) to find f'(x), set f'(x) = 0, find the angles where cos(4x) = -1/2, set up equations for 4x equal to those angles, and solve for x. The x-coordinates are π/12 and 5π/12.

Step-by-step explanation:

To find the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, π], we need to differentiate f(x) and set the resulting equation equal to 0. Differentiating f(x) will give us f'(x), which represents the slope of the function at any given point. Setting f'(x) = 0 will help us find the x-coordinates where the slope is 0. Let's solve this step by step:

1. Differentiate f(x) to find f'(x): f'(x) = 2 + 4cos(4x).
2. Set f'(x) = 0:

2 + 4cos(4x) = 0

4cos(4x) = -2

cos(4x) = -1/2

1. Find the angles where cos(4x) = -1/2. Using the unit circle or trigonometric identities, we can determine that these angles are π/3 and 5π/3.
2. Set up equations for 4x equal to π/3 and 5π/3:

4x = π/3

4x = 5π/3

1. Solve for x:

x = π/12 and x = 5π/12.

Therefore, the x-coordinates where f'(x) = 0 for f(x) = 2x + sin(4x) in the interval [0, π] are π/12 and 5π/12.