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Question
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
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Solution
\[\text{ Let } f\left( x \right) = 3 \cos x + 4 \sin x + 5\]
\[\text{ We know that }\]
\[ - \sqrt{3^2 + 4^2} \leq 3 \cos x + 4 \sin x \leq \sqrt{3^2 + 4^2}\]
\[ \Rightarrow - 5 \leq 3 \cos x + 4 \sin x \leq 5\]
\[ \Rightarrow - 5 + 5 \leq 3 \cos x + 4 \sin x + 5 \leq 5 + 5\]
\[ \Rightarrow 0 \leq f(x) \leq 10\]
\[\text{ Hence, maximum and minimum vales of f(x) are 0 and 10 respectively } .\]
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