Trig function bounded on interval (without calculus), prove that x32sin⁡x+9−x3cos⁡x≤3

Dayton Burnett

Dayton Burnett

Answered

2022-01-29

Trig function bounded on interval (without calculus), prove that x32sinx+9x3cosx3

Answer & Explanation

tipoule137p

tipoule137p

Expert

2022-01-30Added 10 answers

By Cauchy Schwarz
9=(x3+(9x3))(sin2x+cos2x)(x32sinx+9x3cosx)2
and the result follows
vasselefa

vasselefa

Expert

2022-01-31Added 9 answers

For all 0x323 (which is π2)
x32sinx+(9x3)12cosx=
=(x32)2+9x3(x32(x32)2+9x3sinx+(9x3)12(x32)2+9x3cosx)=
=3(x323sinx+(9x3)123cosx)=3sin(x+arccosx323)3

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