For the function f(x) = sin(x) + cos(x), find the x-coordinate of the maximum point in the interval [0, 2π].

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Q: For the function f(x) = sin(x) + cos(x), find the x-coordinate of the maximum point in the interval [0, 2π].

Solution: f'(x) = cos(x) - sin(x). Setting f'(x) = 0 gives tan(x) = 1, so x = π/4 + nπ. In [0, 2π], the maximum occurs at x = 3π/4.

Steps: 9

Step 1: Identify the function we are working with, which is f(x) = sin(x) + cos(x).
Step 2: Find the derivative of the function, f'(x). The derivative is f'(x) = cos(x) - sin(x).
Step 3: Set the derivative equal to zero to find critical points: cos(x) - sin(x) = 0.
Step 4: Rearrange the equation to get tan(x) = 1.
Step 5: Solve for x. The general solution for tan(x) = 1 is x = π/4 + nπ, where n is any integer.
Step 6: Determine the specific solutions in the interval [0, 2π]. For n = 0, x = π/4; for n = 1, x = π/4 + π = 5π/4.
Step 7: Evaluate the function f(x) at the critical points x = π/4 and x = 5π/4 to find which gives the maximum value.
Step 8: Calculate f(π/4) and f(5π/4). Compare the values to find the maximum.
Step 9: Identify that the maximum occurs at x = 3π/4, which is the x-coordinate of the maximum point in the interval [0, 2π].