Increasing and descreasing function question

[appplejack]

Homework Statement

I've got two questions
1.Find the intervals on which f increases and and the intervals on which f decreases.

f(x)= x-cosx, 0≤x≤2∏

2. Why is f(x) = -2sin2x - 2sinx = -2sinx (2cosx + 1)
How can you factor 2cosx from the function on the left side?

Homework Equations

The Attempt at a Solution

1.I get f '(x) = 1 + sinx, 0≤x≤2∏
The answer says that it increases on [0,2∏] but I don't get it.

 

[tiny-tim]

hi appplejack!

2. Why is f(x) = -2sin2x - 2sinx = -2sinx (2cosx + 1)
How can you factor 2cosx from the function on the left side?

learn your trigonometric identities …

sin2x = 2sinxcosx ​

1.I get f '(x) = 1 + sinx, 0≤x≤2∏
The answer says that it increases on [0,2∏] but I don't get it.

1 + sinx is always ≥ 0, isn't it?

 

[appplejack]

Thanks for your help but I think I need to work on trig. Could you explain why 1 + sinx is always ≥ 0?

 

[tiny-tim]

draw it!

 

[appplejack]

I'm weak on trig. For 1 + sinx, does it shift the graph upward by 1 because y intercept is 1
when x=0? And the graph of sine goes up and down. That's why I doubt that the function increases all the time. I know that I'm wrong but I'm still wondering why.

 

[tiny-tim]

And the graph of sine goes up and down. That's why I doubt that the function increases all the time.

1 + sinx is the derivative, it doesn't matter if it goes up and down, so long as it doesn't become negative