Question

The function $f(x)=|x^{2}-3x+2|,x\in\mathbb{R}$ is not differentiable at ________.

• A. $x=1$ and $x=3$
• B. $x=1$ and $x=2$
• C. $x=2$ and $x=4$
• D. $x=4$ and $x=5$
• E. $x=-1$ and $x=-2$

Hint:

Absolute value functions of polynomials are non-differentiable at their simple roots.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The function \(g(x) = |h(x)|\) is generally not differentiable at the points where \(h(x) = 0\), provided that \(h'(x) \neq 0\) at those points. This is because the graph of \(|h(x)|\) has a sharp corner at the roots of \(h(x)\) where it touches and crosses the x-axis.
Step 2: Key Formula or Approach
1. Identify the inner function \(h(x) = x^2 - 3x + 2\).
2. Find the roots of the equation \(h(x) = 0\). These are the potential points of non-differentiability.
3. Check the derivative of the inner function, \(h'(x)\), at these roots. If \(h'(x) \neq 0\), then \(f(x)\) is not differentiable at that root.
Step 3: Detailed Explanation
1. Find the roots of the inner function.
We need to solve \(x^2 - 3x + 2 = 0\).
This quadratic equation can be factored:
\[ (x-1)(x-2) = 0 \] The roots are \(x=1\) and \(x=2\).
These are the points where the function \(f(x)\) might not be differentiable.
2. Check the derivative of the inner function at these roots.
The inner function is \(h(x) = x^2 - 3x + 2\).
Its derivative is \(h'(x) = 2x - 3\).
- At \(x=1\): \(h'(1) = 2(1) - 3 = -1\). Since \(h'(1) \neq 0\), the graph of \(h(x)\) crosses the x-axis at a non-zero slope, creating a sharp corner for \(|h(x)|\). Thus, \(f(x)\) is not differentiable at \(x=1\).
- At \(x=2\): \(h'(2) = 2(2) - 3 = 1\). Since \(h'(2) \neq 0\), the graph of \(h(x)\) also crosses the x-axis at a non-zero slope, creating another sharp corner. Thus, \(f(x)\) is not differentiable at \(x=2\).
3. Conclusion.
The function \(f(x) = |x^2 - 3x + 2|\) is not differentiable at \(x=1\) and \(x=2\).
Step 4: Final Answer
The points of non-differentiability are x = 1 and x = 2.