Is cos &#x2061;<!-- ⁡ --> ( x ) &#x2265;<!-- ≥ --> 1 &#x2212;<!-- − -->

Mylee Underwood

Mylee Underwood

Answered question

2022-07-04

Is cos ( x ) 1 x 2 2 for all x in R?

Answer & Explanation

jugf5

jugf5

Beginner2022-07-05Added 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?