The function f(x) = sin(x) + cos(x) has a maximum value at which point? (2022)

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Question: The function f(x) = sin(x) + cos(x) has a maximum value at which point? (2022)

Options:

Correct Answer: π/4

Exam Year: 2022

Solution:

To find the maximum, set f\'(x) = cos(x) - sin(x) = 0. This occurs at x = π/4.

The function f(x) = sin(x) + cos(x) has a maximum value at which point? (2022)

The function f(x) = sin(x) + cos(x) has a maximum value at which point? (2022)

Step 1: Write down the function f(x) = sin(x) + cos(x).
Step 2: Find the derivative of the function, which 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: Recognize that cos(x) = sin(x) at specific angles, particularly x = π/4.
Step 6: Check if this point is a maximum by evaluating the second derivative or using the first derivative test.

Differentiation – Understanding how to find the derivative of a function to locate critical points.
Trigonometric Functions – Knowledge of the properties and values of sine and cosine functions.
Maxima and Minima – Identifying maximum and minimum values of functions using first derivative tests.