For the function f(x) = sin(x) + cos(x) defined on the interval [0, 2π], which of the following is true about its behavior?

It is increasing on the interval [π/4, 5π/4].
It is decreasing on the interval [π/4, 5π/4].
It is increasing on the intervals [0, π/4] and [5π/4, 7π/4].
It is decreasing on the intervals [0, π/4] and [π/2, 2π].

Correct answer: It is increasing on the interval [π/4, 5π/4].

Solution

The function f(x) = sin(x) + cos(x) is increasing on the interval [π/4, 5π/4] because the derivatives of sin(x) and cos(x) are positive and negative, respectively, in this interval, resulting in a net positive derivative.

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