What is the maximum value of `3cosx+4sinx+5`?

Options

A. 5
B. 7
C. 10
D. 12

Explanation

To find the maximum value of the expression \(3\cos x + 4\sin x + 5\), we can follow these steps:

### Step 1: Identify the coefficients
In the expression \(3\cos x + 4\sin x\), we identify the coefficients:
- \(A = 3\)
- \(B = 4\)

### Step 2: Calculate the maximum value of \(A\cos x + B\sin x\)
The maximum value of the expression \(A\cos x + B\sin x\) can be calculated using the formula:
\[ \text{Maximum value} = \sqrt{A^2 + B^2} \]
Substituting the values of \(A\) and \(B\):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Add the constant term
Now, we need to add the constant term \(5\) from the original expression:
\[ \text{Maximum value of } (3\cos x + 4\sin x + 5) = 5 + 5 = 10 \]

### Conclusion
Thus, the maximum value of \(3\cos x + 4\sin x + 5\) is:
\[ \boxed{10} \]