Discussion
The function \(f(x) = |x^{2} - 3x + 2|, x \in \mathbb{R}\) is not differentiable at
Answer: B
It is not differentiable at \(x = 1\) and \(x = 2\).
\(x^{2} - 3x + 2 = (x - 1)(x - 2)\). The modulus of a function has a corner wherever the function crosses zero with a non-zero slope. Here the derivative \(2x - 3\) is −1 at \(x = 1\) and 1 at \(x = 2\), so the graph folds sharply at both roots and the left and right derivatives differ.
At every other point, including 3, 4, 5, −1 and −2, the quadratic is non-zero and \(|f|\) is smooth.
A quick way to see it is to sketch the parabola and reflect the part below the x-axis upwards; the reflected arc meets the rest of the curve in sharp corners at the two roots.
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