If `f (x) =2x, g (x) =3 sin x -x cos x, then ` for `x in (0, (pi)/(2)):`

Options

A. `f (x) gt g (x)`
B. `f (x) lt g (x)`
C. `f (x)=g (x)` has exactly one real root.
D. `f (x) =g (x)` has exactly two real roots

Answer

To determine which function is greater between \( f(x) = 2x \) and \( g(x) = 3\sin x - x\cos x \) for \( x \) in the interval \( (0, \frac{\pi}{2}) \), we will analyze the derivatives of both functions.

### Step 1: Define the functions
We have:
\[ f(x) = 2x \]
\[ g(x) = 3\sin x - x\cos x \]

### Step 2: Differentiate both functions
Now, we differentiate both functions with respect to \( x \).

1. The derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(2x) = 2 \]

2. The derivative of \( g(x) \):
Using the product rule for the term \( -x\cos x \):
\[ g'(x) = \frac{d}{dx}(3\sin x) - \frac{d}{dx}(x\cos x) \]
\[ = 3\cos x - \left( \cos x + x(-\sin x) \right) \]
\[ = 3\cos x - \cos x + x\sin x \]
\[ = 2\cos x + x\sin x \]

### Step 3: Compare the derivatives
Now we compare \( f'(x) \) and \( g'(x) \):
\[ f'(x) = 2 \]
\[ g'(x) = 2\cos x + x\sin x \]

### Step 4: Analyze the behavior of \( g'(x) \)
For \( x \) in the interval \( (0, \frac{\pi}{2}) \):
- \( \cos x \) is positive and decreases from 1 to 0.
- \( \sin x \) is positive and increases from 0 to 1.

Thus, both \( 2\cos x \) and \( x\sin x \) are positive in this interval. Therefore, \( g'(x) \) is greater than \( f'(x) \):
\[ g'(x) = 2\cos x + x\sin x > 2 \]

### Step 5: Conclusion about the functions
Since \( g'(x) > f'(x) \) for \( x \in (0, \frac{\pi}{2}) \), it implies that \( g(x) \) is increasing faster than \( f(x) \).

To determine which function is greater, we can evaluate at a specific point in the interval, for example, at \( x = 0 \):
\[ f(0) = 2(0) = 0 \]
\[ g(0) = 3\sin(0) - 0\cos(0) = 0 \]

Since both functions are equal at \( x = 0 \) and \( g'(x) > f'(x) \) in the interval, we can conclude that:
\[ g(x) > f(x) \text{ for } x \in (0, \frac{\pi}{2}) \]

### Final Answer
Thus, for \( x \in (0, \frac{\pi}{2}) \), \( g(x) > f(x) \).

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