The function `f(x)=x^(2)-6x+9` is increasing for `x gt3.`

Answer

To determine whether the function \( f(x) = x^2 - 6x + 9 \) is increasing for \( x > 3 \), we can follow these steps:

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first need to find its derivative \( f'(x) \).

\[ f(x) = x^2 - 6x + 9 \]

Using the power rule of differentiation, we differentiate \( f(x) \):

\[ f'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(6x) + \frac{d}{dx}(9) \]
\[ f'(x) = 2x - 6 \]

### Step 2: Set the derivative greater than or equal to zero
Next, we need to determine where the derivative is greater than or equal to zero to find the intervals where the function is increasing.

\[ f'(x) \geq 0 \]
\[ 2x - 6 \geq 0 \]

### Step 3: Solve the inequality
Now, we solve the inequality for \( x \):

\[ 2x \geq 6 \]
\[ x \geq 3 \]

### Step 4: Conclusion
The function \( f(x) \) is increasing for \( x \geq 3 \). Therefore, the statement that the function is increasing for \( x > 3 \) is **true**.