Consider the function
`f(x)=|x^(2)-5x+6|`
What is f'(4) equal to ?

Options

A. `-4`
B. `-3`
C. 3
D. 2

Answer

To find \( f'(4) \) for the function \( f(x) = |x^2 - 5x + 6| \), we will follow these steps:

### Step 1: Identify the function inside the absolute value
The function inside the absolute value is \( g(x) = x^2 - 5x + 6 \).

### Step 2: Factor the quadratic expression
We can factor \( g(x) \):
\[ g(x) = x^2 - 5x + 6 = (x-2)(x-3) \]
This means \( g(x) = 0 \) at \( x = 2 \) and \( x = 3 \).

### Step 3: Determine the intervals for the absolute value
The critical points divide the x-axis into intervals:
1. \( (-\infty, 2) \)
2. \( [2, 3] \)
3. \( (3, \infty) \)

We need to determine the sign of \( g(x) \) in each interval:
- For \( x < 2 \), \( g(x) > 0 \) (e.g., at \( x = 0 \), \( g(0) = 6 \)).
- For \( 2 < x < 3 \), \( g(x) < 0 \) (e.g., at \( x = 2.5 \), \( g(2.5) = -0.25 \)).
- For \( x > 3 \), \( g(x) > 0 \) (e.g., at \( x = 4 \), \( g(4) = 2 \)).

Thus, we can express \( f(x) \) based on these intervals:
- For \( x < 2 \) and \( x > 3 \): \( f(x) = g(x) = x^2 - 5x + 6 \)
- For \( 2 \leq x \leq 3 \): \( f(x) = -g(x) = - (x^2 - 5x + 6) = -x^2 + 5x - 6 \)

### Step 4: Find the derivative \( f'(x) \)
Now we differentiate \( f(x) \) in each interval:
- For \( x < 2 \) and \( x > 3 \):
\[ f'(x) = g'(x) = 2x - 5 \]
- For \( 2 < x < 3 \):
\[ f'(x) = -g'(x) = - (2x - 5) = -2x + 5 \]

### Step 5: Evaluate \( f'(4) \)
Since \( 4 > 3 \), we use the derivative for \( x > 3 \):
\[ f'(4) = 2(4) - 5 = 8 - 5 = 3 \]

### Final Answer
Thus, \( f'(4) = 3 \).

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