Consuder the function
`f(x)=|x^(2)-5x+6|`
What is f'(2.5) equal to ?

Options

A. `-3`
B. `-2`
C. 0
D. 2

Answer

To find \( f'(2.5) \) for the function \( f(x) = |x^2 - 5x + 6| \), we will follow these steps:

### Step 1: Simplify the function inside the modulus
First, we need to factor the quadratic expression inside the modulus:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]
This means that the expression \( x^2 - 5x + 6 \) is equal to zero at \( x = 2 \) and \( x = 3 \).

### Step 2: Determine the intervals for the modulus
The quadratic \( x^2 - 5x + 6 \) is a parabola that opens upwards. It will be negative between its roots (2 and 3) and positive outside this interval. Therefore, we can analyze the sign of the function in the following intervals:
- For \( x < 2 \): \( x^2 - 5x + 6 > 0 \)
- For \( 2 < x < 3 \): \( x^2 - 5x + 6 < 0 \)
- For \( x > 3 \): \( x^2 - 5x + 6 > 0 \)

### Step 3: Rewrite the function based on the intervals
Since we are interested in \( f(x) \) for \( x = 2.5 \) (which is between 2 and 3), we have:
\[ f(x) = -(x^2 - 5x + 6) = -x^2 + 5x - 6 \]

### Step 4: Differentiate the function
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(-x^2 + 5x - 6) = -2x + 5 \]

### Step 5: Evaluate the derivative at \( x = 2.5 \)
Now we substitute \( x = 2.5 \) into the derivative:
\[ f'(2.5) = -2(2.5) + 5 = -5 + 5 = 0 \]

### Conclusion
Thus, the value of \( f'(2.5) \) is:
\[ \boxed{0} \]

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