Find the Maximum and Minimum Values of 3 cos x + 4 sin x + 5
Question:
\[
3\cos x+4\sin x+5
\]
Write its maximum and minimum values.
Solution
\[ a\cos x+b\sin x \] has maximum value \[ \sqrt{a^2+b^2} \] and minimum value \[ -\sqrt{a^2+b^2} \]
Here, \[ a=3,\qquad b=4 \]
\[ \sqrt{3^2+4^2} = \sqrt{9+16} = 5 \]
\[ -5 \leq 3\cos x+4\sin x \leq 5 \]
Adding 5 throughout, \[ 0 \leq 3\cos x+4\sin x+5 \leq 10 \]
Maximum value \[ \boxed{10} \] Minimum value \[ \boxed{0} \]