If `0ltxlt(pi)/(2)`, then the minimum value of `2(sinx+cosx+cosec 2x)^3 ` is

Options

A. (a) 27
B. (b) 13.5
C. (c) 6.75
D. (d) 0

Answer

To find the minimum value of \( 2(\sin x + \cos x + \csc 2x)^3 \) for \( 0 < x < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Understand the expression
We need to minimize the expression \( 2(\sin x + \cos x + \csc 2x)^3 \).

### Step 2: Use the relationship for \(\csc 2x\)
Recall that \( \csc 2x = \frac{1}{\sin 2x} \) and \( \sin 2x = 2 \sin x \cos x \). Therefore, we can rewrite \(\csc 2x\) as:
\[ \csc 2x = \frac{1}{2 \sin x \cos x} \]

### Step 3: Substitute \(\csc 2x\) into the expression
Now, substitute this back into the original expression:
\[ \sin x + \cos x + \csc 2x = \sin x + \cos x + \frac{1}{2 \sin x \cos x} \]

### Step 4: Apply the Arithmetic Mean-Geometric Mean (AM-GM) Inequality
We can apply the AM-GM inequality to the three terms \(\sin x\), \(\cos x\), and \(\frac{1}{2 \sin x \cos x}\):
\[ \frac{\sin x + \cos x + \frac{1}{2 \sin x \cos x}}{3} \geq \sqrt[3]{\sin x \cos x \cdot \frac{1}{2 \sin x \cos x}} = \sqrt[3]{\frac{1}{2}} \]

### Step 5: Simplify the inequality
This simplifies to:
\[ \sin x + \cos x + \frac{1}{2 \sin x \cos x} \geq 3 \sqrt[3]{\frac{1}{2}} \]

### Step 6: Cube both sides
Cubing both sides gives:
\[ \left(\sin x + \cos x + \frac{1}{2 \sin x \cos x}\right)^3 \geq \left(3 \sqrt[3]{\frac{1}{2}}\right)^3 = 27 \cdot \frac{1}{2} = \frac{27}{2} \]

### Step 7: Multiply by 2
Now, we multiply the entire inequality by 2:
\[ 2\left(\sin x + \cos x + \csc 2x\right)^3 \geq 27 \]

### Step 8: Conclusion
Thus, the minimum value of \( 2(\sin x + \cos x + \csc 2x)^3 \) is \( 27 \).

### Final Answer
The minimum value is \( \boxed{27} \).

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