What will be the maximum value of `15-(x-3)^(2)` ?

Answer

To find the maximum value of the expression \( 15 - (x - 3)^2 \), we can follow these steps:

### Step 1: Understand the expression
The expression \( 15 - (x - 3)^2 \) consists of a constant (15) and a quadratic term \(-(x - 3)^2\). The term \(-(x - 3)^2\) represents a downward-opening parabola.

### Step 2: Identify the vertex of the parabola
The vertex of the parabola given by \( -(x - 3)^2 \) occurs at \( x = 3 \). This is because the expression \( (x - 3)^2 \) reaches its minimum value (which is 0) when \( x = 3 \).

### Step 3: Substitute \( x = 3 \) into the expression
Now, we will substitute \( x = 3 \) into the expression to find the maximum value:
\[ 15 - (3 - 3)^2 = 15 - 0 = 15 \]

### Step 4: Conclusion
Thus, the maximum value of the expression \( 15 - (x - 3)^2 \) is \( 15 \).