 In this video, we provide the solution to question number six from the practice final exam for math 1210, we're asked to find the points x for which the graph f of x equals x plus sine of x has a horizontal tangent line, where in is any integer where which you'll see in the answers, there's gonna be some ends and that's because there's got to be multiple horizontal tangent lines. Well, if you have a horizontal tangent line, that means that your derivative is equal to zero so we need to compute the derivative f and figure out where it's equal to zero. So by usual derivative rules, the derivative of f of x here would be one plus cosine of x, we have to see when that equals zero. Subtracting one, we get that cosine of x is equal to negative one. So we're trying to figure out where cosine is equal to negative one. If we think of the usual unit circle diagram, cosine be the x coordinate negative one would be over here. So we're looking at multiples of pi, but we'll only want odd multiples of pi because if we take like zero or two pi four pi is not going to work here, because those actually would make cosine equal to one. So we need to have odd multiples of pi, in which case that then leads us to choosing choice D x needs to be pi plus two pi n. So if you take any integer in, if you add any multiple of two pi to pi, well, that's positive or negative, that always will give you the odd multiples of pi. And so that then gives us the correct answer D.