Find the Maximum and Minimum Value of 12 cos x + 5 sin x + 4

Solution

We know that the expression

\[ a\cos x + b\sin x \]

has maximum value

\[ \sqrt{a^2+b^2} \]

and minimum value

\[ -\sqrt{a^2+b^2} \]

Given expression:

\[ 12\cos x + 5\sin x + 4 \]

Here,

\[ a=12,\qquad b=5 \]

Now,

\[ \sqrt{a^2+b^2} = \sqrt{12^2+5^2} \]

\[ = \sqrt{144+25} \]

\[ = \sqrt{169} \]

\[ =13 \]

Therefore,

Maximum value of \[ 12\cos x + 5\sin x \] is \[ 13 \]

So, maximum value of the given expression is

\[ 13+4=17 \]

Minimum value of \[ 12\cos x + 5\sin x \] is \[ -13 \]

So, minimum value of the given expression is

\[ -13+4=-9 \]

Final Answer

\[ \boxed{\text{Maximum value }=17} \]

\[ \boxed{\text{Minimum value }=-9} \]