Maximize 2sinxcosx/[(1+sinx)(1+cosx)]

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• anemone
In summary, the purpose of "Maximize 2sinxcosx/[(1+sinx)(1+cosx)]" is to find the maximum value of the expression. To solve it, you can use calculus techniques such as taking the derivative and setting it equal to zero to find critical values, and then using the second derivative test to determine if these critical values correspond to a maximum value. The critical values for this equation are x = 0 and x = π/2, and the maximum value is 1, which occurs at x = π/4. This occurs because at x = π/4, the numerator is equal to the denominator, resulting in a maximum value of 1, and the second derivative at
anemone
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MHB
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Maximize $\dfrac{2\sin x \cos x}{(1+\sin x)(1+\cos x)}$ for $x\in \left(0, \dfrac{\pi}{2}\right)$.

AM-GM inequality

\begin{align*}\dfrac{2\sin x \cos x}{(1+\sin x)(1+\cos x)}&=\dfrac{2\sin x \cos x (1-\sin x)}{(1-\sin^2 x)(1+2\cos^2 \dfrac{x}{2}-1)}\\&=\dfrac{2\sin x \cos x (1-\sin x)}{(\cos^2 x)(2\cos^2 \dfrac{x}{2})}\\&=\dfrac{\sin x}{\cos x}\left(\dfrac{1-2\sin \dfrac{x}{2} \cos \dfrac{x}{2}}{\cos^2 \dfrac{x}{2}}\right)\\&=\tan x\left(\sec^2 \dfrac{x}{2}-2\tan \dfrac{x}{2}\right)\\&=\dfrac{2\tan \dfrac{x}{2}}{1-\tan^2 \dfrac{x}{2}}\left(1+\tan^2 \dfrac{x}{2}-2\tan \dfrac{x}{2}\right)\\&=2\tan \dfrac{x}{2}\dfrac{\left(1-\tan \dfrac{x}{2}\right)\left(1-\tan \dfrac{x}{2}\right)}{\left(1+\tan \dfrac{x}{2}\right)\left(1-\tan \dfrac{x}{2}\right)}\\&=2\tan \dfrac{x}{2}\left(\dfrac{\tan \dfrac{\pi}{4}-\tan \dfrac{x}{2}}{1+\tan \dfrac{\pi}{2}\tan \dfrac{x}{2}}\right)\\&=2\tan \dfrac{x}{2}\tan \left( \dfrac{\pi}{4}-\dfrac{x}{2}\right)\end{align*}

Therefore by the AM-GM inequality,

\begin{align*}\sqrt{\dfrac{2\sin x \cos x}{(1+\sin x)(1+\cos x)}}&=\sqrt{2\tan \dfrac{x}{2}\tan \left( \dfrac{\pi}{4}-\dfrac{x}{2}\right)}\le \dfrac{\sqrt{2}}{2}\left(\tan \dfrac{\pi}{2}+\tan \left( \dfrac{\pi}{4}-\dfrac{x}{2}\right)\right)\end{align*}

Equality attains when $\tan \dfrac{\pi}{2}=\tan \left( \dfrac{\pi}{4}-\dfrac{x}{2}\right)$, i.e. when $x=\dfrac{\pi}{4}$ where $\tan \dfrac{\pi}{8}=\sqrt{2}-1$, an exact value we could get from using the double angle formula for $\tan x$ and that $\tan \dfrac{\pi}{4}=1$.

Hence $\dfrac{2\sin x \cos x}{(1+\sin x)(1+\cos x)}\le 2(\sqrt{2}-1)^2=2(3-2\sqrt{2})$.

1. What is the formula for "Maximize 2sinxcosx/[(1+sinx)(1+cosx)]"?

The formula for "Maximize 2sinxcosx/[(1+sinx)(1+cosx)]" is a trigonometric identity known as the double angle formula for sine, which states that sin2x = 2sinxcosx. This can be rearranged to get the given expression.

2. How do you find the maximum value of this expression?

To find the maximum value of the expression, we can use the properties of trigonometric functions and calculus. First, we can rewrite the expression as 2sinxcosx/(1+sinx)(1+cosx) = 2sinxcosx/(1+sinxcosx+sinx+cosx). Then, we can use the first derivative test to find the critical points, and plug them into the second derivative to determine whether they are maximum or minimum points. The maximum value can then be found by evaluating the expression at the critical points and choosing the largest value.

3. Can this expression be simplified?

Yes, this expression can be simplified by using the trigonometric identity for the sum of two angles, which states that sin(x+y) = sinxcosy + cosxsiny. By applying this identity, the expression can be rewritten as 2sinxcosx/(1+sinx)(1+cosx) = 2sinxcosx/(1+sinx+cosx+sinxcosx) = 2sinxcosx/(2+sinxcosx). This simplification may make it easier to find the maximum value.

4. What is the domain of this expression?

The domain of this expression is all real numbers except for values that would make the denominator equal to 0. This includes values of x that would make sinxcosx = -2, which is not possible for real numbers. Therefore, the domain is all real numbers except for x = (2n+1)π/2, where n is any integer.

5. Can this expression be used to solve real-world problems?

While this expression may not have direct applications in real-world problems, it is a useful tool in mathematics and science for understanding trigonometric identities and solving related problems. It can also be used in engineering and physics for calculating maximum values of certain functions. However, it is not typically used in everyday situations.

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