Let \(f(x) = |x^{2} - 3x|\). The left-hand derivative and the right-hand derivative of f at \(x = 3\) are, respectively,
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −3 and 3
Since \(x^{2} - 3x = x(x - 3)\), the expression inside is negative for 0 < x < 3 and positive for x > 3. Removing the modulus:
\(0 \lt x \lt 3: \ f(x) = 3x - x^{2}\)
\(x \gt 3: \ f(x) = x^{2} - 3x\)
Both pieces are 0 at x = 3, so the one-sided derivatives are the derivatives of the pieces at 3:
\(Lf'(3) = 3 - 2(3) = -3\)
\(Rf'(3) = 2(3) - 3 = 3\)
So they are −3 and 3 (unequal, so f is not differentiable at 3).
Why the other options are incorrect
Option (1) 3 and −3 — Incorrect. These are the right-hand and left-hand derivatives in the wrong order; on the left the function is falling to 0.
Option (3) −1 and 1 — Incorrect. These are the slopes for |x − 3|; here the inside x(x − 3) has slope 3 at x = 3, not 1.
Option (4) 0 and 3 — Incorrect. The left piece \(3x - x^{2}\) has slope −3 at x = 3, not 0.