For what value of x does the function y = x + 2 cos x have a horizontal tangent line?
I said f'(x) = 1 + 0(cosx) + 2(-sinx) = 1 - 2sinx
So if the line is horizontal, f'(x) must = 0, so 0 = 1 - 2sinx --> sin x = 1/2
Now I am stuck. What value of x should I put? 30 degrees? pi divided by six?
It may be asking for any value of x that satisfies the condition. If so, then you can list whichever solution you want.
It may ask for the value(s) that occurs in a particular range (i.e. between 0 and pi). In this case you should find them all in that range.
Or, it may ask you to list EVERY value of x that would work. For example, recalling trig, there are two solutions between 0 and 2pi:
But it is periodic, so we could list the solutions like so:
or, probably better:
where n is an integer
However it is sort of ambiguous as written so you might want to ask your teacher what he/she expects you to answer when it is worded like this. We can only guess if the problem is not worded more explicitly.