The maximum of the function 3cos x–4sin x is :

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the maximum of the function \( f(x) = 3\cos x - 4\sin x \), we can use the concept that any function of the form \( a\cos x + b\sin x \) can be expressed in terms of a single cosine function with a phase shift. Here's how we can solve it step by step:

### Step 1: Identify coefficients
We have the function:
\[ f(x) = 3\cos x - 4\sin x \]
Here, \( a = 3 \) and \( b = -4 \).

### Step 2: Calculate the magnitude
To find the maximum value of \( f(x) \), we need to calculate the magnitude \( R \) of the vector formed by the coefficients \( a \) and \( b \):
\[ R = \sqrt{a^2 + b^2} \]
Substituting the values:
\[ R = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Determine the range of the function
The function \( f(x) = a\cos x + b\sin x \) will oscillate between \( -R \) and \( R \). Therefore, the range of \( f(x) \) is:
\[ -R \leq f(x) \leq R \]
This means:
\[ -5 \leq f(x) \leq 5 \]

### Step 4: Find the maximum value
From the range we calculated, the maximum value of \( f(x) \) is:
\[ \text{Maximum value} = R = 5 \]

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

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