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Mylee Underwood 2022-07-04 Answered
Is cos ( x ) 1 x 2 2 for all x in R?
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Answers (1)

jugf5
Answered 2022-07-05 Author has 18 answers
Consider the function
f ( x ) = cos x 1 + x 2 2
We have f(0)=0 and
f ( x ) = x sin x
with f′(0)=0. Also
f ( x ) = 1 cos x
which shows that f′(x) is a strictly increasing function, because its derivative is positive except on a set of isolated points (that has no limit point). Therefore f′(x)>0 for x>0 and f′(x)<0 for x<0. Hence 0 is an absolute minimum for f. Since f(0)=0, we have f(x)>0 for every x 0. This means that, for every x,
cos x 1 x 2 2
equality holding only for x=0.

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