Section 3.3 Examples
Calculus I
Andy Long

#39, p. 222

f[x_]:=2Cos[x]+Cos[x]^2

a) We need to study the sign of the derivative to get intervals of increase and decrease:

Simplify[f'[x]]
(* Let's start by thinking about what Sine and Cosine look like: *)
Plot[{Sin[x],Cos[x]},{x,0,2Pi}]

Since (1+Cos[x]) is always greater than or equal to zero, and zero at only one point (Pi), we have a critical point, but look to the rest of the function (-2 Sin[x]) to tell us the sign of f’[x]. So negative from 0 to Pi, and positive from Pi to 2Pi.

Simplify[f''[x]]
Solve[f''[x]==0,x]

Plot[f[x],{x,0,2Pi}]
Plot[f'[x],{x,0,2Pi}]
Plot[f''[x],{x,0,2Pi}]