The maximum value of `cos xsin x+sqrt(sin^(2)x+sin^(2)((pi)/(6))}` is

Options

A. `sqrt(5)/(3)`
B. `sqrt((3)/(2))`
C. `sqrt((5)/(2))`
D. `sqrt(5)/(2)`

Explanation

To find the maximum value of the expression \( y = \cos x \sin x + \sqrt{\sin^2 x + \sin^2 \left( \frac{\pi}{6} \right)} \), we can follow these steps:

### Step 1: Simplify the expression
We know that \( \sin^2 \left( \frac{\pi}{6} \right) = \frac{1}{4} \). Therefore, we can rewrite the expression as:
\[ y = \cos x \sin x + \sqrt{\sin^2 x + \frac{1}{4}} \]

### Step 2: Rewrite the expression
We can express \( y \) in a more manageable form:
\[ y = \cos x \sin x + \sqrt{\sin^2 x + \frac{1}{4}} = \cos x \sin x + \sqrt{\sin^2 x + \sin^2 \left( \frac{\pi}{6} \right)} \]

### Step 3: Factor out common terms
Let’s factor out \( \cos x \):
\[ y = \cos x \left( \sin x + \sqrt{\frac{1}{4} + \sin^2 x} \right) \]

### Step 4: Find the maximum value
To maximize \( y \), we need to analyze the term \( \sin x + \sqrt{\sin^2 x + \frac{1}{4}} \).

### Step 5: Use the Cauchy-Schwarz inequality
Using the Cauchy-Schwarz inequality, we can find that:
\[ \left( \sin x + \sqrt{\sin^2 x + \frac{1}{4}} \right)^2 \leq (1 + 1) \left( \sin^2 x + \frac{1}{4} \right) \]
This simplifies to:
\[ \left( \sin x + \sqrt{\sin^2 x + \frac{1}{4}} \right)^2 \leq 2 \left( \sin^2 x + \frac{1}{4} \right) \]

### Step 6: Set up the discriminant
To find the maximum value, we can also set up a quadratic equation in terms of \( t = \tan x \):
\[ y^2 \tan^2 x - 2y \tan x + \frac{y^2}{4} = 0 \]
The discriminant must be non-negative for real values of \( \tan x \):
\[ (-2y)^2 - 4 \cdot y^2 \cdot \frac{y^2}{4} \geq 0 \]
This simplifies to:
\[ 4y^2 - y^4 \geq 0 \]

### Step 7: Solve the inequality
Factoring gives:
\[ y^2(4 - y^2) \geq 0 \]
This implies:
\[ y^2 \leq 4 \quad \Rightarrow \quad |y| \leq 2 \]

### Step 8: Find the maximum value
Thus, the maximum value of \( y \) is:
\[ y \leq 2 \]

### Conclusion
The maximum value of the original expression \( \cos x \sin x + \sqrt{\sin^2 x + \sin^2 \left( \frac{\pi}{6} \right)} \) is \( \sqrt{5} \).

### Final Answer
The maximum value of the given expression is \( \frac{\sqrt{5}}{2} \).

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