The maximum value of `f(x)=(sin2x)/(sinx+cosx)` in the interval `(0, (pi)/(2))` is

Options

A. `sqrt2`
B. `(1)/(sqrt2)`
C. 1
D. `(1)/(2)`

Answer

To find the maximum value of the function \( f(x) = \frac{\sin 2x}{\sin x + \cos x} \) in the interval \( (0, \frac{\pi}{2}) \), we will follow these steps:

### Step 1: Differentiate the Function
To find the maximum value, we first need to differentiate \( f(x) \) with respect to \( x \).

Using the quotient rule:
\[ f'(x) = \frac{(\sin x + \cos x)(\cos 2x) - \sin 2x(\cos x - \sin x)}{(\sin x + \cos x)^2} \]

### Step 2: Set the Derivative to Zero
Next, we set the numerator of \( f'(x) \) equal to zero to find critical points:
\[ (\sin x + \cos x)\cos 2x - \sin 2x(\cos x - \sin x) = 0 \]

### Step 3: Simplify the Equation
We can simplify the equation:
\[ \sin x \cos 2x + \cos x \cos 2x - \sin 2x \cos x + \sin 2x \sin x = 0 \]

### Step 4: Use Trigonometric Identities
Using the identity \( \sin 2x = 2 \sin x \cos x \), we can rewrite the equation:
\[ \sin x \cos 2x + \cos x \cos 2x - 2 \sin x \cos x \cos x + 2 \sin x \cos x \sin x = 0 \]

### Step 5: Factor the Equation
This can be factored to find solutions for \( x \):
\[ \cos 2x(\sin x + \cos x) = 2 \sin x \cos x \]

### Step 6: Solve for Critical Points
We can solve for \( x \) by finding where \( \cos 2x = 0 \) or \( \sin x + \cos x = 0 \) or \( 2 \sin x \cos x = 0 \).

The critical point occurs at:
\[ \sin 2x = 1 \implies 2x = \frac{\pi}{2} \implies x = \frac{\pi}{4} \]

### Step 7: Evaluate the Function at the Critical Point
Now we evaluate \( f(x) \) at \( x = \frac{\pi}{4} \):
\[ f\left(\frac{\pi}{4}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}} = \frac{1}{\frac{2}{\sqrt{2}}} = \frac{\sqrt{2}}{2} \]

### Step 8: Check the Endpoints
Since we are looking for the maximum value in the interval \( (0, \frac{\pi}{2}) \), we should also check the limits as \( x \) approaches \( 0 \) and \( \frac{\pi}{2} \):
- As \( x \to 0 \), \( f(0) = 0 \).
- As \( x \to \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 0 \).

### Conclusion
The maximum value of \( f(x) \) in the interval \( (0, \frac{\pi}{2}) \) is:
\[ \boxed{1} \]