Direction: For the next two (2) items that follow: Consider the function f(x) = |x 2– 5x + 6|
What is f’’(2.5) equal to?
-2
Finding the Second Derivative of an Absolute Value Function
The function given is \(f(x) = |x^2 - 5x + 6|\). We need to find the value of the second derivative, \(f''(x)\), at \(x = 2.5\).
First, let's analyze the expression inside the absolute value: \(g(x) = x^2 - 5x + 6\). To understand the absolute value function, we need to know where \(g(x)\) is positive, negative, or zero.
We find the roots of the quadratic equation \(x^2 - 5x + 6 = 0\):
\((x - 2)(x - 3) = 0\)
The roots are \(x = 2\) and \(x = 3\). These roots divide the number line into three intervals: \( (-\infty, 2) \), \( (2, 3) \), and \( (3, \infty) \).
We can determine the sign of \(x^2 - 5x + 6\) in each interval:
- For \(x < 2\), pick \(x=0\): \(0^2 - 5(0) + 6 = 6 > 0\). So, \(x^2 - 5x + 6\) is positive.
- For \(2 < x < 3\), pick \(x=2.5\): \((2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 < 0\). So, \(x^2 - 5x + 6\) is negative.
- For \(x > 3\), pick \(x=4\): \(4^2 - 5(4) + 6 = 16 - 20 + 6 = 2 > 0\). So, \(x^2 - 5x + 6\) is positive.
Based on the sign of \(x^2 - 5x + 6\), we can write the piecewise definition of \(f(x) = |x^2 - 5x + 6|\):
- If \(x \le 2\) or \(x \ge 3\), \(f(x) = x^2 - 5x + 6\)
- If \(2 < x < 3\), \(f(x) = -(x^2 - 5x + 6) = -x^2 + 5x - 6\)
Calculating the Derivative at x = 2.5
We need to find \(f''(2.5)\). The point \(x = 2.5\) lies in the interval \( (2, 3) \). In this interval, the function is defined as \(f(x) = -x^2 + 5x - 6\).
Let's find the first derivative \(f'(x)\) for \(x\) in the interval \( (2, 3) \):
\(f'(x) = \frac{d}{dx}(-x^2 + 5x - 6)\)
\(f'(x) = -2x + 5\)
Now, let's find the second derivative \(f''(x)\) for \(x\) in the interval \( (2, 3) \):
\(f''(x) = \frac{d}{dx}(-2x + 5)\)
\(f''(x) = -2\)
Since \(x = 2.5\) is in the interval \( (2, 3) \), we use the expression for \(f''(x)\) derived for this interval. In this interval, \(f''(x)\) is a constant value of \(-2\).
Therefore, \(f''(2.5) = -2\).
The second derivative of \(f(x) = |x^2 - 5x + 6|\) at \(x = 2.5\) is \(-2\).
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