Write the maximum and minimum values of 3 cosx + 4 sinx + 5.

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Solution

Let f(x)=3 cosx + 4 sinx + 5
We know that,
$- \sqrt{3^{2} + 4^{2}} \leq 3 c o s x + 4 s i n x \leq \sqrt{3^{2} + 4^{2}}$
$\Rightarrow - \sqrt{9 + 16} \leq 3 c o s x + 4 s i n x \leq \sqrt{9 + 16}$
$\Rightarrow - \sqrt{25} \leq 3 c o s x + 4 s i n x \leq \sqrt{25}$
$\Rightarrow - 5 \leq 3 c o s x + 4 s i n x \leq 5$
$\Rightarrow - 5 + 5 \leq 3 c o s x + 4 s i n x + 5 \leq 5 + 5$
$\Rightarrow 0 \leq 3 c o s x + 4 s i n x + 5 \leq 10$ Hence, minimum and maximum values of 3 cosx + 4 sinx +5 are 0 and 10 respectively.

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