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 $f''(x)$

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

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

2.  when $4\cos x$ the graph of the function is concave downward and this is for

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