What is the maximum value of the function y = min `(12-x, 8+x)`?

Options

A. 12
B. 10
C. 11
D. 8

Answer

To find the maximum value of the function \( y = \min(12 - x, 8 + x) \), we can follow these steps:

### Step 1: Set the two expressions equal to each other
We start by equating the two expressions inside the minimum function:
\[ 12 - x = 8 + x \]

### Step 2: Solve for \( x \)
Now, we will solve the equation for \( x \):
\[ 12 - 8 = x + x \]
\[ 4 = 2x \]
\[ x = \frac{4}{2} = 2 \]

### Step 3: Evaluate \( y \) at \( x = 2 \)
Next, we will substitute \( x = 2 \) back into both expressions to find the value of \( y \):
1. For \( 12 - x \):
\[ 12 - 2 = 10 \]
2. For \( 8 + x \):
\[ 8 + 2 = 10 \]

### Step 4: Determine the maximum value of \( y \)
Since both expressions are equal at \( x = 2 \) and equal to 10, we find:
\[ y = \min(10, 10) = 10 \]

### Conclusion
Thus, the maximum value of the function \( y = \min(12 - x, 8 + x) \) is:
\[ \boxed{10} \]

---