The maximum value of `12 sin theta -9 sin ^(2) theta ` is

Options

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

Answer

To find the maximum value of the expression \( 12 \sin \theta - 9 \sin^2 \theta \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( x = \sin \theta \). The expression can be rewritten as:
\[ f(x) = 12x - 9x^2 \]
where \( -1 \leq x \leq 1 \).

### Step 2: Find the derivative
To find the maximum value, we first need to find the critical points by taking the derivative of \( f(x) \):
\[ f'(x) = 12 - 18x \]

### Step 3: Set the derivative to zero
Setting the derivative equal to zero to find critical points:
\[ 12 - 18x = 0 \]
\[ 18x = 12 \]
\[ x = \frac{12}{18} = \frac{2}{3} \]

### Step 4: Evaluate the function at critical points and endpoints
Now we evaluate \( f(x) \) at the critical point \( x = \frac{2}{3} \) and at the endpoints \( x = -1 \) and \( x = 1 \).

1. **At \( x = \frac{2}{3} \)**:
\[ f\left(\frac{2}{3}\right) = 12\left(\frac{2}{3}\right) - 9\left(\frac{2}{3}\right)^2 \]
\[ = 8 - 9 \cdot \frac{4}{9} = 8 - 4 = 4 \]

2. **At \( x = -1 \)**:
\[ f(-1) = 12(-1) - 9(-1)^2 = -12 - 9 = -21 \]

3. **At \( x = 1 \)**:
\[ f(1) = 12(1) - 9(1)^2 = 12 - 9 = 3 \]

### Step 5: Determine the maximum value
Now we compare the values obtained:
- \( f\left(\frac{2}{3}\right) = 4 \)
- \( f(-1) = -21 \)
- \( f(1) = 3 \)

The maximum value is:
\[ \boxed{4} \]