The sum of maximum and minimum values of the expression `5 cosx + 3sin(pi/6 -x)+4` is

Answer

To find the sum of the maximum and minimum values of the expression \(5 \cos x + 3 \sin\left(\frac{\pi}{6} - x\right) + 4\), we can follow these steps:

### Step 1: Simplify the Expression
Let \(f(x) = 5 \cos x + 3 \sin\left(\frac{\pi}{6} - x\right) + 4\).

Using the sine subtraction formula, we have:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
Here, \(a = \frac{\pi}{6}\) and \(b = x\).

Thus,
\[ \sin\left(\frac{\pi}{6} - x\right) = \sin\left(\frac{\pi}{6}\right) \cos x - \cos\left(\frac{\pi}{6}\right) \sin x \]
Substituting the values, we get:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
So,
\[ \sin\left(\frac{\pi}{6} - x\right) = \frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \]

### Step 2: Substitute Back into the Expression
Now substituting back into \(f(x)\):
\[ f(x) = 5 \cos x + 3\left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right) + 4 \]
This simplifies to:
\[ f(x) = 5 \cos x + \frac{3}{2} \cos x - \frac{3\sqrt{3}}{2} \sin x + 4 \]
Combining like terms:
\[ f(x) = \left(5 + \frac{3}{2}\right) \cos x - \frac{3\sqrt{3}}{2} \sin x + 4 \]
\[ f(x) = \frac{13}{2} \cos x - \frac{3\sqrt{3}}{2} \sin x + 4 \]

### Step 3: Identify the Coefficients
Let \(A = \frac{13}{2}\) and \(B = -\frac{3\sqrt{3}}{2}\).

### Step 4: Find Maximum and Minimum Values
The expression \(A \cos x + B \sin x\) can be bounded by:
\[ -\sqrt{A^2 + B^2} \leq A \cos x + B \sin x \leq \sqrt{A^2 + B^2} \]
Calculating \(A^2 + B^2\):
\[ A^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4}, \quad B^2 = \left(-\frac{3\sqrt{3}}{2}\right)^2 = \frac{27}{4} \]
Thus,
\[ A^2 + B^2 = \frac{169}{4} + \frac{27}{4} = \frac{196}{4} = 49 \]
So,
\[ \sqrt{A^2 + B^2} = \sqrt{49} = 7 \]

### Step 5: Determine the Range of \(f(x)\)
Thus,
\[ -7 \leq \frac{13}{2} \cos x - \frac{3\sqrt{3}}{2} \sin x \leq 7 \]
Adding 4 to all parts of the inequality:
\[ -7 + 4 \leq f(x) \leq 7 + 4 \]
This gives:
\[ -3 \leq f(x) \leq 11 \]

### Step 6: Find the Sum of Maximum and Minimum Values
The maximum value of \(f(x)\) is \(11\) and the minimum value is \(-3\).
Thus, the sum of the maximum and minimum values is:
\[ 11 + (-3) = 8 \]

### Final Answer
The sum of the maximum and minimum values of the expression is \(8\).

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