Today I was studying this function $\displaystyle{y=\sin \left|x \right|+\left|\cos x \right|}$ and I tried to draw its graph using my knowledge so I started from the domain which is $∀x \in\mathbb R$ , then it is clear the fact that the function is even.
Then I have to find the intersection's point with $x=0$ and $y=0$, here I don't know what to do, I'm sure that there is an intersection with $x=0$ in the point $C(0,1)$.
Then it is important to know when the function is positive or negative so $\displaystyle{y>0}$, I would say that it is always positive but it is wrong. After this, it is fundamental to find the maxima and the minima of the function and to do this I have to derive the function and calculate when the derivative is positive $\displaystyle{y'>0}$, but also here I have no clear idea of what I have to do.. Can someone clear my ideas and show me what should I do? (Excuse me if I've not been clear)
Thank you in advance!!!
Here are some hints to get you started.
First, focus attention on values $x \geq 0$, since, as you said, the function is even. So $\sin |x|$ can always be replaced with $\sin x$ on this interval.
Next, note that the function is $2\pi$-periodic on $[0, +\infty)$. So you can start by studying the interval $[0,2\pi]$.
Then study the four intervals $[0,\pi/2]$, $[\pi/2,\pi]$, $[\pi,3\pi/2]$, $[3\pi/2,2\pi]$, separately, as $|\cos x|$ can be replaced with either $\cos x$ or $-\cos x$ on each interval.
For example, to solve an equation $\sin x = -\cos x$, you can divide both sides by $\cos x$ (after first verifying that $\cos x \ne 0$ if the equation is true).