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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