Increasing and Decreasing

y = sinx + cosx

y' = cosx - sinx

Setting to 0, we have

cosx - sin x = 0    add sin x to both sides

cosx = sinx         and this is true at  pi/4 and (5/4)pi on [0, 2pi]

And taking the second derivative, we have

y" = -sin x - cosx

So....evaluating this at pi/4, we have

-sin(pi/4) - cos(pi/4) = -(1/√2√2) - (1/√2) = -2/√2 ....so the curve is concave down at tthis point

And evaluating the second derivative at  (5/4)pi, we have

-sin(5/4 pi) - cos(5/4pi) =  (1/√2√2) - (-1/√2) = 2/√2 ....so the curve is concave up at this point

So....on [0, 2pi],   the graph would be increasing on [0, pi/4) ....and it would be decreasing on (pi/4, 5/4 pi)..... and then it would be increasing again at (5/4 pi, 2pi]......and it would repeat this behavior on every interval  [n(2pi), (n+1)(2pi)]

Here's the graph......https://www.desmos.com/calculator/twbniyyrfu

Nice , thank you chris