If f(x) = sin(x) + cos(x), then the critical points in the interval [0, 2π] are:

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Question: If f(x) = sin(x) + cos(x), then the critical points in the interval [0, 2π] are:

Options:

Correct Answer: π/4, 5π/4

Solution:

To find critical points, we set f\'(x) = cos(x) - sin(x) = 0. This gives tan(x) = 1, leading to x = π/4 and x = 5π/4 in the interval [0, 2π].

If f(x) = sin(x) + cos(x), then the critical points in the interval [0, 2π] are:

If f(x) = sin(x) + cos(x), then the critical points in the interval [0, 2π] are:

Step 1: Identify the function 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 cos(x) = sin(x).
Step 5: Divide both sides by cos(x) (assuming cos(x) is not zero) to get tan(x) = 1.
Step 6: Find the angles where tan(x) = 1. These angles are x = π/4 and x = 5π/4.
Step 7: Check if these angles are within the interval [0, 2π]. Both π/4 and 5π/4 are in this interval.

Finding Critical Points – This involves taking the derivative of a function and setting it to zero to find points where the function's slope is zero.
Trigonometric Functions – Understanding the behavior of sine and cosine functions, particularly their values and relationships at specific angles.
Interval Analysis – Identifying critical points within a specified interval, ensuring that solutions fall within the given bounds.