What is the Interval for Increasing Function y=x+sinxcosx?

[Kamataat]

Find the interval in which the function $y=x+\sin x\cos x$ is increasing. So, first I differentiated to get $y'=1+\cos 2x$. Then I set $y'$ equal to zero:

$$1+\cos 2x=0$$
$$\cos 2x=-1$$
$$2x=\pm \arccos m+2n\pi$$, where $$n\in\mathbb{Z}$$
$$2x=\pm \arccos(-1)+2n\pi$$
$$2x=\pm\pi+2n\pi$$
$$x=\pm\frac{\pi}{2}+n\pi$$

So, since $y'=0$ is true only at certain points (because $n\in\mathbb{Z}$), we know that the function is strictly increasing or decreasing. To find out which, we do this:

$$x_1=-50 : y(x_1)=y_1=-49.75$$
$$x_2=30 : y(x_2)=y_2=29.85$$

Thus the function is strictly increasing because in the case of $x_1 < x_2$ we have $y_1 < y_2$.

So the function is increasing on the open interval $X^{\uparrow}=]-\infty;\infty[$.

Is this correct?

- Kamataat

 

[dextercioby]

True,the derivative is never negative...But you can't say that the function's increasing on R,you'll have to exclude the infinite amount of points where it is zero...

Daniel.

 

[Kamataat]

So it's strictly increasing on $X^\uparrow=\mathbb{R}\ \backslash\ (\pm\pi/2+\pi n)$, where $n\in\mathbb{Z}$?

- Kamataat

 

[dextercioby]

Remove the minus,if the "n" is integer,automatically the value $-\frac{\pi}{2}$ is assumed.

Daniel.

 

[Kamataat]

ok, but why is -pi/2 assumed if n is an integer? is there a mathematical reason or is it just agreed on by mathematicians?

- Kamataat

 

[dextercioby]

Take "n=-1" in the "plus" formula...

Daniel.

 

[Kamataat]

I fail to see why this should create a problem .

- Kamataat

 

[dextercioby]

Because mathematicians like to count the DISTINCT elements of a set ONLY ONCE...

Daniel.

 

[Kamataat]

yeah, i know that. but i don't get why i have to assume it's a "minus" and not a "plus". wouldn't n=1 in the "minus" formula cause the same problem as n=-1 in the "plus" formula?

- Kamataat

 

[arildno]

You can of course use the minus version rather then the plus version. But, as Daniel said, to use both is redundant (and hence, not well liked).
Stick to the one you like best (most would prefer the plus version).

 

[Kamataat]

ok, that's how i understand it. i just got confused, because i thought he meant to use only the "minus" version. thanks anyway, both of you!

- Kamataat