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Question
Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\] lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]
Solution
\[\text{ Let } f\left( x \right) = (2\sqrt{3} + 3) \sin x + 2\sqrt{3}\cos x\]
\[\text{ We know that }, \]
\[ - \sqrt{\left( 2\sqrt{3} + 3 \right)^2 + \left( 2\sqrt{3} \right)^2} \leq f\left( x \right) \leq \sqrt{\left( 2\sqrt{3} + 3 \right)^2 + \left( 2\sqrt{3} \right)^2}\]
\[ \Rightarrow - \sqrt{12 + 9 + 12\sqrt{3} + 12} \leq f\left( x \right) \leq \sqrt{12 + 9 + 12\sqrt{3} + 12}\]
\[ \Rightarrow - \sqrt{33 + 12\sqrt{3}} \leq f\left( x \right) \leq \sqrt{33 + 12\sqrt{3}}\]
\[\text{ Disclaimer } : \text{ Instead of }- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right),\text{ it should be } - \sqrt{33 + 12\sqrt{3}}\text{ and } \sqrt{33 + 12\sqrt{3}} .\]
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