Question

For what values of x in [0,2π) does f(x)=sin(x)+cos(x) attain its maximum value?

a) x= π/2
b) x= π/4
c) x= 3π/4
d) x=π

Answer 1

The maximum value of f(x) = sin(x) + cos(x) in the interval [0, 2π) is 1, and it occurs at x = π/4 and x = 5π/4.

Step-by-step explanation:

To find the values of x in [0,2π) where f(x) = sin(x) + cos(x) attains its maximum value, we need to determine the critical points of the function.

The maximum value occurs at points where the derivative of the function is equal to 0 or does not exist.

Differentiating f(x), we get f'(x) = cos(x) - sin(x).

Setting f'(x) equal to 0, we have cos(x) = sin(x), which occurs at x = π/4 and x = 5π/4 within the given interval.

Before checking whether these points are maximum or minimum values, we also need to check the endpoints of the interval.

Evaluating f(0), we get f(0) = sin(0) + cos(0) = 0 + 1 = 1. Evaluating f(2π), we get f(2π) = sin(2π) + cos(2π) = 0 + 1 = 1.

Comparing these values with the values at the critical points, we see that the maximum value of f(x) is 1 and it occurs at the critical points x = π/4 and x = 5π/4.