Quadratic trigonometric maximum

Geometry Level 2

Find the maximum value of 4 sin ⁡ 2 x − 12 sin ⁡ x + 7 4\sin^{2} x-12\sin x+7 .

See this nice problem .
None of these choices 25 15 23 19

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Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

By observation, we note that when sin ⁡ x = − 1 \sin{x} = -1 , the expression is maximum:

4 ( − 1 ) 2 − 12 ( − 1 ) + 7 = 4 + 12 + 7 = 23 4(-1)^2-12(-1)+7 = 4+12+7 = \boxed{23}

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that's why its level 1 problem sir

Tanishq Varshney - 6 years, 2 months ago

Could you Help me with observation?

Swetank Raj - 6 years, 1 month ago

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We know that sin ⁡ x ∈ [ − 1 , 1 ] \sin{x} \in [-1,1] . The term 4 sin ⁡ 2 x 4\sin^2{x} is always positive and has the largest value when sin ⁡ x = ± 1 \sin{x} = \pm 1 . The term − 12 sin ⁡ x -12\sin{x} has the largest value when x = − 1 x = -1 . And 7 7 is a constant. Therefore, the largest value of 4 sin ⁡ 2 x − 12 sin ⁡ x + 7 4\sin^2{x}-12\sin{x}+7 is when x = − 1 x=-1 .

Chew-Seong Cheong - 6 years, 1 month ago

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Thank you!

Swetank Raj - 6 years, 1 month ago

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