`f(x) = cos(x) + (1/2)cos(2x), 0<=x<=2pi` (a) Use a graph of `f` to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of `f''` to give better estimates.
See Graph Blue colour represents second derivative and black color represents function.
Rough estimates from graph
Concave down in the intervals (0,pi/3) , (5pi/6,4pi/3) and (5pi/3,2pi)
Concave up in the interval (pi/3,5pi/6) , (4pi/3,5pi/3)
Inflection points at x=0.9 , x=2.6 , x=3.7 and x=5.35
Here is the graph
It looks like the graph is concave down from `(0,1/2pi)` and `(pi,4.1)` . The graph seems concave up from `(1/2pi,pi)` and `(4.1,2pi)`
Find the second derivative:
Take the derivative again
Concave down from `(0,0.9)` and `(2.5,3.75)` and `(5.3, 2pi)`
Concave up from `(0.9,2.5)` and `(3.75, 5.3)`