The maximum value of the expression `sin theta cos^(2)theta(AA theta in [0, pi])` is

Options

A. `(2)/(3)`
B. `(2)/(sqrt3)`
C. `(2)/(3sqrt3)`
D. `(1)/(sqrt3)`

Answer

To find the maximum value of the expression \( f(\theta) = \sin \theta \cos^2 \theta \) for \( \theta \in [0, \pi] \), we will follow these steps:

### Step 1: Define the function
Let \( f(\theta) = \sin \theta \cos^2 \theta \).

### Step 2: Differentiate the function
To find the maximum value, we need to differentiate \( f(\theta) \) with respect to \( \theta \):
\[ f'(\theta) = \frac{d}{d\theta} (\sin \theta \cos^2 \theta) \]
Using the product rule, where \( u = \sin \theta \) and \( v = \cos^2 \theta \):
\[ f'(\theta) = u'v + uv' \]
Calculating \( u' \) and \( v' \):
- \( u' = \cos \theta \)
- \( v = \cos^2 \theta \) and \( v' = 2 \cos \theta (-\sin \theta) = -2 \cos \theta \sin \theta \)

Thus,
\[ f'(\theta) = \cos \theta \cos^2 \theta - 2 \sin^2 \theta \cos \theta \]
Factoring out \( \cos \theta \):
\[ f'(\theta) = \cos \theta (\cos^2 \theta - 2 \sin^2 \theta) \]

### Step 3: Set the derivative to zero
To find critical points, set \( f'(\theta) = 0 \):
\[ \cos \theta (\cos^2 \theta - 2 \sin^2 \theta) = 0 \]
This gives us two cases:
1. \( \cos \theta = 0 \)
2. \( \cos^2 \theta - 2 \sin^2 \theta = 0 \)

### Step 4: Solve for \( \theta \)
**Case 1:** \( \cos \theta = 0 \)
- This occurs at \( \theta = \frac{\pi}{2} \).

**Case 2:** \( \cos^2 \theta - 2 \sin^2 \theta = 0 \)
- Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can rewrite this as:
\[ \cos^2 \theta = 2(1 - \cos^2 \theta) \]
\[ 3 \cos^2 \theta = 2 \implies \cos^2 \theta = \frac{2}{3} \]
Thus, \( \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{2}{3} = \frac{1}{3} \).

### Step 5: Find corresponding \( \theta \)
From \( \sin^2 \theta = \frac{1}{3} \):
\[ \sin \theta = \frac{1}{\sqrt{3}} \quad (\text{since } \theta \in [0, \pi]) \]
Using the Pythagorean identity to find \( \cos \theta \):
\[ \cos \theta = \sqrt{\frac{2}{3}} \quad (\text{since } \cos \theta \geq 0 \text{ in } [0, \frac{\pi}{2}]) \]

### Step 6: Calculate maximum value
Now we can calculate \( f(\theta) \) for both critical points:
1. For \( \theta = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) \cos^2\left(\frac{\pi}{2}\right) = 1 \cdot 0 = 0 \]

2. For \( \sin \theta = \frac{1}{\sqrt{3}} \) and \( \cos^2 \theta = \frac{2}{3} \):
\[ f(\theta) = \sin \theta \cos^2 \theta = \frac{1}{\sqrt{3}} \cdot \frac{2}{3} = \frac{2}{3\sqrt{3}} = \frac{2\sqrt{3}}{9} \]

### Conclusion
Thus, the maximum value of \( f(\theta) = \sin \theta \cos^2 \theta \) for \( \theta \in [0, \pi] \) is:
\[ \boxed{\frac{2\sqrt{3}}{9}} \]