Find the intervals on which f is increasing or decreasing and find the local maximum and minimum values of f. Find the intervals of concavity and the inflection points. f(x)=sinx+cosx, 0

Hello! This function is infinitely differential on entire real line, therefore we can use derivatives to determine its behavior.
Note that f(x) = sinx + cosx = sqrt(2) sin(x+pi/4).
The first derivative is f'(x) = sqrt(2) cos(x + pi/4). It is zero (inside [0, 2 pi] ) at x_1 = pi/4 and x_2 = pi/4 + pi = (5pi)/4.
This way the first derivative is positive (and f(x) increases) on (0, pi/4) and on ((5pi)/4, 2pi) and is negative (and f(x) decreases) on (pi/4, (5pi)/4). Thus f(x) has a local maximum at x_1 = pi/4 with the value f(pi/4) = sqrt(2) and the local minimum at x_2=(5pi)/4 with the value f(x_2)=-sqrt(2).
The second derivative is f''(x) = -sqrt(2) sin(x+pi/4), and it is negative on (0, (3pi)/4) and on ((7pi)/4,2pi) and is positive on ((3pi)/4,(7pi)/4). Therefore f(x) is concave down on (0, (3pi)/4) and on ((7pi)/4,2pi) and is concave up on ((3pi)/4,(7pi)/4).
The inflection points are where direction of concavity changes, i.e. (3pi)/4 and (7pi)/4.
Look at the picture made with desmos.com. The function is in green, the first derivative is dashed in red and the second derivative is dotted in red.