The range of the function `f(x)=sqrt(3-x)+sqrt(2+x)` is :

Options

A. `[2sqrt(2),sqrt(11)]`
B. `[sqrt(5),sqrt(10)]`
C. `[sqrt(5),sqrt(13)]`
D. `[sqrt(2),sqrt(7])`

Answer

To find the range of the function \( f(x) = \sqrt{3 - x} + \sqrt{2 + x} \), follow these steps:

### Step 1: Determine the Domain
The function involves square roots, so the expressions inside the square roots must be non-negative.

1. For \( \sqrt{3 - x} \):
\[ 3 - x \geq 0 \implies x \leq 3 \]

2. For \( \sqrt{2 + x} \):
\[ 2 + x \geq 0 \implies x \geq -2 \]

Combining these inequalities, the domain of \( f(x) \) is:
\[ -2 \leq x \leq 3 \]

### Step 2: Find Critical Points
To find the critical points, we need to differentiate \( f(x) \) and set the derivative equal to zero.

1. Differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx} \left( \sqrt{3 - x} + \sqrt{2 + x} \right) \]
Using the chain rule:
\[ f'(x) = \frac{-1}{2\sqrt{3 - x}} + \frac{1}{2\sqrt{2 + x}} \]

2. Set \( f'(x) = 0 \):
\[ \frac{-1}{2\sqrt{3 - x}} + \frac{1}{2\sqrt{2 + x}} = 0 \]
Simplifying:
\[ \frac{1}{\sqrt{2 + x}} = \frac{1}{\sqrt{3 - x}} \]
Squaring both sides:
\[ 2 + x = 3 - x \]
Solving for \( x \):
\[ 2x = 1 \implies x = \frac{1}{2} \]

### Step 3: Evaluate \( f(x) \) at Critical Points and Endpoints
Evaluate \( f(x) \) at \( x = -2 \), \( x = \frac{1}{2} \), and \( x = 3 \).

1. At \( x = -2 \):
\[ f(-2) = \sqrt{3 - (-2)} + \sqrt{2 + (-2)} = \sqrt{5} + \sqrt{0} = \sqrt{5} \]

2. At \( x = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = \sqrt{3 - \frac{1}{2}} + \sqrt{2 + \frac{1}{2}} = \sqrt{\frac{5}{2}} + \sqrt{\frac{5}{2}} = 2\sqrt{\frac{5}{2}} = \sqrt{10} \]

3. At \( x = 3 \):
\[ f(3) = \sqrt{3 - 3} + \sqrt{2 + 3} = \sqrt{0} + \sqrt{5} = \sqrt{5} \]

### Step 4: Determine the Range
From the evaluations, the minimum value of \( f(x) \) is \( \sqrt{5} \) and the maximum value is \( \sqrt{10} \). Therefore, the range of \( f(x) \) is:
\[ [\sqrt{5}, \sqrt{10}] \]

### Final Answer
The range of the function \( f(x) = \sqrt{3 - x} + \sqrt{2 + x} \) is:
\[ \boxed{[\sqrt{5}, \sqrt{10}]} \]