SineCosine at its pinnacle

Geometry Level 2

Let the maximum value of sin ⁡ x cos ⁡ x \sin x \cos x for x ∈ R x \in \mathbb{R} be M M . If the value of x x for which sin ⁡ x cos ⁡ x \sin x \cos x is maximum, where 0 < x < π 0< x < \pi , can be expressed as A π B \dfrac{A\pi}{B} for positive coprime integers A A and B B , compute M + A + B M+A+B .


The answer is 5.5.

Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

Nihar Mahajan
Sep 6, 2015

sin ⁡ x cos ⁡ x = 2 sin ⁡ x cos ⁡ x 2 = sin ⁡ 2 x 2 ≤ 1 2 = 0.5 \sin x \cos x \\ = \dfrac{2\sin x \cos x}{2} \\ = \dfrac{\sin 2x}{2} \\ \leq \dfrac{1}{2} = \Large\boxed{0.5}

And this maximum is attained when sin ⁡ 2 x = 1 ⇒ 2 x = π 2 ⇒ x = π 4 \sin 2x = 1 \Rightarrow 2x = \dfrac{\pi}{2} \Rightarrow\boxed{ x=\dfrac{\pi}{4}} .

Thus M = 0.5 , A = 1 , B = 4 ⇒ M + A + B = 0.5 + 1 + 4 = 5.5 M=0.5 \ , \ A = 1 \ , \ B = 4 \Rightarrow M+A+B = 0.5+1+4=\huge\boxed{5.5}

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So simple Liked your solution and upvoted!

Department 8 - 5 years, 9 months ago

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Thanks bro!

Nihar Mahajan - 5 years, 9 months ago

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