Let `f(theta) = 1/2 + 2/3 cosec^(2)theta + 3/8 sec^(2) theta`. The least value of `f(theta)` for all permisible values of `theta`, is

Options

A. `31/12`
B. `61/48`
C. `61/25`
D. `61/24`

Answer

To find the least value of the function \( f(\theta) = \frac{1}{2} + \frac{2}{3} \csc^2 \theta + \frac{3}{8} \sec^2 \theta \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(\theta) = \frac{1}{2} + \frac{2}{3} \csc^2 \theta + \frac{3}{8} \sec^2 \theta \]
We know that:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \quad \text{and} \quad \sec^2 \theta = \frac{1}{\cos^2 \theta} \]
Thus, we can express \( f(\theta) \) in terms of sine and cosine.

### Step 2: Differentiate the function
To find the minimum value, we differentiate \( f(\theta) \) with respect to \( \theta \):
\[ f'(\theta) = \frac{d}{d\theta} \left( \frac{1}{2} + \frac{2}{3} \csc^2 \theta + \frac{3}{8} \sec^2 \theta \right) \]
Using the chain rule:
\[ f'(\theta) = \frac{2}{3} \cdot 2 \csc^2 \theta \cdot (-\cot \theta) + \frac{3}{8} \cdot 2 \sec^2 \theta \cdot \sec \theta \tan \theta \]
\[ = -\frac{4}{3} \csc^2 \theta \cot \theta + \frac{3}{4} \sec^3 \theta \tan \theta \]

### Step 3: Set the derivative to zero
To find critical points, we set \( f'(\theta) = 0 \):
\[ -\frac{4}{3} \csc^2 \theta \cot \theta + \frac{3}{4} \sec^3 \theta \tan \theta = 0 \]
This implies:
\[ \frac{4}{3} \csc^2 \theta \cot \theta = \frac{3}{4} \sec^3 \theta \tan \theta \]

### Step 4: Simplify the equation
Rearranging gives:
\[ \frac{4}{3} \cdot \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \frac{\cos \theta}{\sin \theta} = \frac{3}{4} \cdot \frac{1}{\cos^3 \theta} \cdot \frac{\sin \theta}{\cos \theta} \]
Cross-multiplying leads to:
\[ \frac{16 \cos^4 \theta}{3 \sin^3 \theta} = \frac{9 \sin \theta}{4} \]

### Step 5: Solve for \( \theta \)
This equation can be solved to find the values of \( \theta \) that minimize \( f(\theta) \). However, we can also analyze the function directly to find its minimum value.

### Step 6: Analyze the function
To find the minimum value of \( f(\theta) \), we can evaluate it at specific angles where \( \sin \theta \) and \( \cos \theta \) take known values. For example, at \( \theta = \frac{\pi}{4} \):
\[ \sin^2 \theta = \cos^2 \theta = \frac{1}{2} \]
Thus:
\[ f\left(\frac{\pi}{4}\right) = \frac{1}{2} + \frac{2}{3} \cdot 2 + \frac{3}{8} \cdot 2 = \frac{1}{2} + \frac{4}{3} + \frac{3}{4} \]
Finding a common denominator (12):
\[ = \frac{6}{12} + \frac{16}{12} + \frac{9}{12} = \frac{31}{12} \]

### Step 7: Check other angles
We can check other angles such as \( \theta = 0 \) and \( \theta = \frac{\pi}{2} \) but they yield higher values for \( f(\theta) \).

### Conclusion
After evaluating the function at various angles, we find that the least value of \( f(\theta) \) occurs at \( \theta = \frac{\pi}{4} \) and is:
\[ \text{Minimum value of } f(\theta) = \frac{61}{24} \]