A geometry problem by Aly Ahmed

Geometry Level 3

sin ⁡ 2 x sin ⁡ x + 8 \dfrac{\sin^2 x}{\sin x + 8} Find the sum of the maximum value and the minimum value of the expression above for real x x .


The answer is 0.1428.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

Since sin ⁡ 2 x ≥ 0 \sin^2 x\geq 0 and sin ⁡ x + 8 > 0 \sin x+8>0 , the minimum is 0 0 . Maximum will be obtained when the denominator is minimum, which is 8 − 1 = 7 8-1=7 when sin ⁡ x = − 1 \sin x=-1 . Hence the sum is 0 + 1 7 ≈ 0.142857 0+\dfrac {1}{7}\approx \boxed {0.142857} .

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