# `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...

`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.

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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:

`f'(x)=-sin(2x)-sinx`

Take the derivative again

`f''(x)=-2cos(2x)-cosx`

Graph

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)`