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462chevelle
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Homework Statement
Use the intermediate value theorem and rolles theorem to prove that the equation has exactly one real solution
2x-2-cos(x)=0
The Attempt at a Solution
Let the interval be [a,b] and let f(a)<0 and f(b)>0
Then by the IVT there must be at least one zero between a and b.
f'(x)=2+sin(x)
since f'(x) doesn't = 0 anywhere and its always >0, therefore f(x) is increasing throughout its entire domain. Therefore f(a) cannot = f(b) anywhere.
I feel like I am doing a bad job at explaining this, but this is my first proof for class ever, other than geometry in high school and i was bad at it. Is there anything terribly wrong or that could be improved upon at all?
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