Question

F( X)=X+2COSX find intervals of concavity and inflection points 0 14 years ago

Answer 1

The derivatives for the trig functions cos & sin go in the following order:
sin
cos
-sin
-cos
(and back again through those four)

f(x) = x + 2cos(x)
f'(x) = -2sin(x)
f''(x) = -2cos(x)

So the second derivative looks like a standard -cos(x) with twice the magnitude (e.g. starts at -2, reaches 0 at x=pi/2, reaches +2 by x=pi, and so on. So the concavity is down, then up, etc. The points of inflection are at x = pi/2 +/- k*pi, where k is a natural counting integer (1,2,3 ...)