Question

Determine the open intervals on which the graph of f(x) = x - 4cos x is concave upward or concave downward

Answer 1

The graph of the function
`f(x)` is

• concave upward, when
  `f''(x)>0;`
• concave downward, when
  `f''(x)<0.`

Find
`f''(x):`

1.

`f'(x)=(x-4\cos x)'=1+4\sin x;`

2.

`f''(x)=4\cos x.`

Now:

1. when
`4\cos x>0,` the graph of the function is concave upward and this is for

`x\in \left(-(\pi)/(2)+2\pi k,(\pi)/(2)+2\pi k\right), \text{ where } k\in Z.`

2. when
`4\cos x<0,` the graph of the function is concave downward and this is for

`x\in \left((\pi)/(2)+2\pi k,(3\pi)/(2)+2\pi k\right), \text{ where } k\in Z.`