If `0ltxlt(pi)/(2)`, then the minimum value of
`(sinx+cosx+cosec2x)`,is

Options

A. 27
B. `(27)/(2)`
C. `(25)/(4)`
D. none of these

Explanation

To find the minimum value of the expression \( \sin x + \cos x + \csc 2x \) for \( 0 < x < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the expression
The expression we want to minimize is:
\[ f(x) = \sin x + \cos x + \csc 2x \]
We know that \( \csc 2x = \frac{1}{\sin 2x} \) and \( \sin 2x = 2 \sin x \cos x \). Thus, we can rewrite \( \csc 2x \) as:
\[ \csc 2x = \frac{1}{2 \sin x \cos x} \]
So, our function becomes:
\[ f(x) = \sin x + \cos x + \frac{1}{2 \sin x \cos x} \]

### Step 2: Use the AM-GM Inequality
We can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to the terms \( \sin x \), \( \cos x \), and \( \frac{1}{2 \sin x \cos x} \):
\[ \frac{\sin x + \cos x + \frac{1}{2 \sin x \cos x}}{3} \geq \sqrt[3]{\sin x \cos x \cdot \frac{1}{2 \sin x \cos x}} \]
This simplifies to:
\[ \frac{\sin x + \cos x + \frac{1}{2 \sin x \cos x}}{3} \geq \sqrt[3]{\frac{1}{2}} \]
Multiplying both sides by 3 gives:
\[ \sin x + \cos x + \frac{1}{2 \sin x \cos x} \geq 3 \sqrt[3]{\frac{1}{2}} \]

### Step 3: Calculate the minimum value
Now we calculate \( 3 \sqrt[3]{\frac{1}{2}} \):
\[ 3 \sqrt[3]{\frac{1}{2}} = 3 \cdot \frac{1}{\sqrt[3]{2}} = \frac{3}{\sqrt[3]{2}} \approx 3 \cdot 0.7937 \approx 2.381 \]
Thus, we have:
\[ \sin x + \cos x + \csc 2x \geq 3 \sqrt[3]{\frac{1}{2}} \approx 2.381 \]

### Step 4: Conclusion
The minimum value of \( \sin x + \cos x + \csc 2x \) for \( 0 < x < \frac{\pi}{2} \) is approximately \( 2.381 \).