Lower estimate for <msubsup> &#x222B;<!-- ∫ --> 0 T </msubsup> f ( x ) si

Cristopher Knox

Cristopher Knox

Answered question

2022-06-30

Lower estimate for 0 T f ( x ) sin f ( x ) d x

Answer & Explanation

Giovanna Erickson

Giovanna Erickson

Beginner2022-07-01Added 14 answers

Here's a very clumsy lower bound. By the mean value theorem, for x ( 0 , T ) we have for some c ( 0 , x )
f ( x ) = f ( 0 ) + f ( c ) x > f ( 0 ) T ,
the inequality from the fact | f ( c ) | < 1.
Since sin f ( x ) [ 1 , 1 ], the integrand is at least
| f ( 0 ) T |
imputing a lower bound for the integral of
T | f ( 0 ) T | .

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