What is the maximum value of `f(x)=3-(x-2)^(2)` ?

Options

A. `-3`
B. `-1`
C. 1
D. 3

Explanation

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

### Step 1: Understand the function
The function \( f(x) = 3 - (x - 2)^2 \) is a quadratic function in the standard form \( f(x) = a - (x - h)^2 \), where \( a = 3 \) and \( h = 2 \). The term \( (x - 2)^2 \) is always non-negative since it is a square.

### Step 2: Analyze the square term
Since \( (x - 2)^2 \geq 0 \) for all \( x \), we can conclude that:
\[ -(x - 2)^2 \leq 0 \]
This means that the maximum value of \( - (x - 2)^2 \) is \( 0 \), which occurs when \( (x - 2)^2 = 0 \).

### Step 3: Find when the square term is zero
The square term \( (x - 2)^2 = 0 \) when:
\[ x - 2 = 0 \implies x = 2 \]

### Step 4: Substitute back to find \( f(x) \)
Now, substituting \( x = 2 \) back into the function:
\[ f(2) = 3 - (2 - 2)^2 = 3 - 0 = 3 \]

### Step 5: Conclusion
Thus, the maximum value of \( f(x) \) is:
\[ \text{Maximum value} = 3 \]

### Summary
The maximum value of the function \( f(x) = 3 - (x - 2)^2 \) is \( 3 \).

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