2 sinx +2 cosx is minimum velu please get the and the sums

2sinx+2cosx is  minimum velu
please  get  the  and the sums


1 Answers

jagdish singh singh
173 Points
5 years ago
\hspace{-0.70 cm}$ Using $\bf{A.M\geq G.M}$ Inequality for $2^{\sin x}\;,2^{\cos x}>0$ \\\\ So $2^{\sin x}+2^{\cos x}\geq 2\cdot 2^{\frac{\sin x+\cos x}{2}}\geq 2\cdot 2^{-\frac{1}{\sqrt{2}}}=2^{1-\frac{1}{\sqrt{2}}}$\\\\ bcz $-\sqrt{2} \leq \sin x+\cos x \leq \sqrt{2}$ for all $x\in \mathbb{R}$\\\\ So $2^{-\sqrt{2}}\leq 2^{\sin x+\cos x}\leq 2^{\sqrt{2}}\Rightarrow 2^{1-\frac{1}{\sqrt{2}}}\leq 2^{\frac{\sin x+\cos x}{2}}\leq 2^{1+\frac{1}{\sqrt{2}}}$\\\\ So $2^{\sin x}+2^{\cos x}\geq 2^{1-\frac{1}{\sqrt{2}}}$ at $x=2n\pi+\frac{5\pi}{4}\;, n\in \mathbb{Z}$

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