Prove that 3 x &#x2212;<!-- − --> x 3 </msup> &lt; 2

Emmy Dillon 2022-06-24 Answered
Prove that
3 x x 3 < 2 sin 2 x , x ( 0 , π 2 )
I have tried by proving that
3 x x 3 < 9 5 π x + 3 2 < 2 sin 2 x , x ( 0 , π 2 )
with Jensen's inequality, but I hoped this problem would have a simpler solution.
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Answers (2)

humusen6p
Answered 2022-06-25 Author has 22 answers
HINT:
sin 2 x 1 2 sin 2 x 2
and
3 x x 3 2
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flightwingsd2
Answered 2022-06-26 Author has 5 answers
The range of the function f ( x ) = 3 x x 3 for x ( 0 , π / 2 ) is f ( x ) ( 0 , 2 )
For the function g ( x ) = 2 sin 2 x for x ( 0 , π / 2 ) the range is clearly g ( x ) ( 2 , )
The inequality is quite clear now.
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