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

Emmy Dillon

Emmy Dillon

Answered question

2022-06-24

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.

Answer & Explanation

humusen6p

humusen6p

Beginner2022-06-25Added 22 answers

HINT:
sin 2 x 1 2 sin 2 x 2
and
3 x x 3 2
flightwingsd2

flightwingsd2

Beginner2022-06-26Added 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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