Write the maximum and minimum values of
Question:

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.

Solution:

Let $f(x)=3 \cos x+4 \sin x+5$

We know that,

$-\sqrt{3^{2}+4^{2}} \leq 3 \cos x+4 \sin x \leq \sqrt{3^{2}+4^{2}}$

$\Rightarrow-5 \leq 3 \cos x+4 \sin x \leq 5$

$\Rightarrow-5+5 \leq 3 \cos x+4 \sin x+5 \leq 5+5$

$\Rightarrow 0 \leq f(x) \leq 10$

Hence, maximum and minimum vales of $f(x)$ are 0 and 10 respectively.

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