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

• A. 3.5
• B. 4.5
• C. 3
• D. 4

Answer 1

The Correct Option is B

The equation is \( a + 2b = 6 \). To find the maximum and minimum values of \( a + b \), express \( a \) in terms of \( b \): \[ a = 6 - 2b \] The expression for \( a + b \) is: \[ a + b = (6 - 2b) + b = 6 - b \] Since \( a \geq 0 \) and \( b \geq 0 \), we have \( 6 - 2b \geq 0 \) and \( b \geq 0 \).
Therefore, \( b \) can vary between 0 and 3. 
- When \( b = 0 \), \( a + b = 6 \) (maximum value). 
- When \( b = 3 \), \( a + b = 3 \) (minimum value). 
The average of the maximum and minimum values is: \[ \frac{6 + 3}{2} = 4.5 \] To find the average of the maximum and minimum values of an expression, first find the extreme values and then take their average.