Let f(x) = 2x + cos (x). I need to prove their is only one real root. However, I am not given a range. I know I need to find a range on which the sign changes and use the Intermediate Value Theorem, but how do I know my range is correct and am I only looking at the section I chose as the function is periodic and has many roots.

Question

Let f(x) = 2x + cos (x).  I need to prove their is only one real root. However, I am not given a range. I know I need to find a range on which the sign changes and use the Intermediate Value Theorem, but how do I know my range is correct and am I only looking at the section I chose as the function is periodic and has many roots.

anonymous · Answer

well if you can determine that the slope is positive for all values of x, and then determine that there is both a negative value of 2x + cos x, and a positive value of 2x + cos x, then you will be finished.

anonymous · Answer

oh, also assume the function is continuous for all x (which it is)

anonymous · Answer

the IVT is a condition that guarantees a root. but if you know the slope of this function is always positive, that means the function is always increasing... i hope this helps you establish why this condition makes that root unique

anonymous · Answer

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