Question

find the points of inflection and identify where the graph of f(x)=x+2cosx is concave up or concave down on the interval [0,2pi].

find the limit approaching positive infinity of x-cosx/x

Answer 1

\[f(x)=x+2\cos x~~\Rightarrow~~f'(x)=1-2\sin x~~\Rightarrow~~f''(x)=-2\cos x\]
Solve for \(x\) with \(0\le x\le2\pi\):
\[-2\cos x=0\]
You should get two solutions.

For the limit:
\[\lim_{x\to\infty}\frac{x-\cos x}{x}=\frac{\infty}{\infty}\]
Apply L'Hopital's rule:
\[\lim_{x\to\infty}\frac{1-\sin x}{1}=\frac{1-(\pm1)}{1}\]
The oscillating nature says the limit doesn't exist.