If x &#x2208;<!-- ∈ --> ( 0 , &#x03C0;<!-- π --> 4

arridsd9

arridsd9

Answered question

2022-06-06

If x ( 0 , π 4 ) , Then prove that cos x ( sin 2 x ) ( cos x sin x ) > 8

Answer & Explanation

Reagan Madden

Reagan Madden

Beginner2022-06-07Added 15 answers

MySolution:: Using A.M≥G.M
2 sin x ( cos x sin x ) cos x
so
4 sin x ( cos x sin x ) cos 2 x sin 2 x ( cos x sin x ) cos x cos x sin x 4
So
sin 2 x ( cos x sin x ) cos x sin 2 x 8 < 1 8
So
cos x sin 2 x ( cos x sin x ) > 8
Davion Harding

Davion Harding

Beginner2022-06-08Added 4 answers

We are to prove that
cos x sin x ( cos x sin x ) > 8 sin x
By Cauchy-Schwarz Inequality, since all quantities involved are positive
L H S = 1 cos x + 1 cos x sin x 4 sin x + cos x sin x = 4 sin x
For the given range of x, we have 1 > 2 sin 2 x
So
4 sin x > 4 sin x × 2 sin 2 x = 8 sin x
and we are done

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