If $$ a $$ and $$ b $$ are non-negative real numbers such that
$$ a + 2b = 6 $$,
then the average of the maximum and minimum possible values of $$ (a + b) $$ is:

Question

If $$ a $$ and $$ b $$ are non-negative real numbers such that  
$$ a + 2b = 6 $$,  
then the average of the maximum and minimum possible values of $$ (a + b) $$ is:

Accepted Answer

To solve this problem:

1. **Rewrite the Equation**:  
   From the given equation:  
   $$
   a + 2b = 6 \implies a + b = 6 - b.
   $$

2. **Analyze Extreme Cases**:  
   - Since $$ a $$ and $$ b $$ are non-negative real numbers, the value of $$ b $$ can range from **0** to **3** (as $$ 2b \leq 6 \implies b \leq 3$$).  
   - **Maximum Value of $$ a + b $$**:  
     When $$ b = 0 $$, we have:  
     $$
     a + b = 6 - 0 = 6.
     $$  
   - **Minimum Value of $$ a + b $$**:  
     When $$ b = 3 $$, we have:  
     $$
     a + b = 6 - 3 = 3.
     $$

3. **Calculate the Average**:  
   The average of the maximum and minimum values is:  
   $$
   	ext{Average} = \frac{	ext{Maximum} + 	ext{Minimum}}{2} = \frac{6 + 3}{2} = 4.5.
   $$

Thus, the correct answer is **4.5**.

Answer

Answer

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