At x = `(5pi)/(6),` f(x)= 2sin 3x+3 cos 3x is

Options

A. maximum
B. minimum
C. 0
D. none of these

Answer

To determine if the function \( f(x) = 2\sin(3x) + 3\cos(3x) \) at \( x = \frac{5\pi}{6} \) is a maximum, minimum, zero, or none of these, we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}[2\sin(3x) + 3\cos(3x)] \]

Using the chain rule, we differentiate each term:

\[ f'(x) = 2 \cdot 3\cos(3x) - 3 \cdot 3\sin(3x) = 6\cos(3x) - 9\sin(3x) \]

### Step 2: Set the derivative to zero
To find critical points, we set the derivative equal to zero:

\[ 6\cos(3x) - 9\sin(3x) = 0 \]

Rearranging gives:

\[ 6\cos(3x) = 9\sin(3x) \]

Dividing both sides by 3:

\[ 2\cos(3x) = 3\sin(3x) \]

### Step 3: Express in terms of tangent
We can express this in terms of tangent:

\[ \tan(3x) = \frac{2}{3} \]

### Step 4: Substitute \( x = \frac{5\pi}{6} \)
Now we will check if \( x = \frac{5\pi}{6} \) satisfies this equation:

\[ 3x = 3 \cdot \frac{5\pi}{6} = \frac{15\pi}{6} = \frac{5\pi}{2} \]

Now we find \( \tan\left(\frac{5\pi}{2}\right) \):

The angle \( \frac{5\pi}{2} \) corresponds to \( \frac{5\pi}{2} - 2\pi = \frac{\pi}{2} \), where \( \tan\left(\frac{\pi}{2}\right) \) is undefined (or infinity). Thus:

\[ \tan\left(\frac{5\pi}{2}\right) \neq \frac{2}{3} \]

### Step 5: Evaluate \( f\left(\frac{5\pi}{6}\right) \)
Next, we evaluate \( f\left(\frac{5\pi}{6}\right) \):

\[ f\left(\frac{5\pi}{6}\right) = 2\sin\left(3 \cdot \frac{5\pi}{6}\right) + 3\cos\left(3 \cdot \frac{5\pi}{6}\right) \]

Calculating \( 3x = \frac{15\pi}{6} = \frac{5\pi}{2} \):

\[ f\left(\frac{5\pi}{6}\right) = 2\sin\left(\frac{5\pi}{2}\right) + 3\cos\left(\frac{5\pi}{2}\right) \]

Using the values of sine and cosine:

\[ \sin\left(\frac{5\pi}{2}\right) = 1 \quad \text{and} \quad \cos\left(\frac{5\pi}{2}\right) = 0 \]

Thus:

\[ f\left(\frac{5\pi}{6}\right) = 2 \cdot 1 + 3 \cdot 0 = 2 \]

### Conclusion
Since \( f\left(\frac{5\pi}{6}\right) = 2 \), it is neither a maximum, minimum, nor zero. Therefore, the answer is **none of these**.

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