Based on the following given information answer the question
Consider the function \(f(x) = x|x|\), \(x \in \mathrm{R}\).
Consider the following statements for the curve \(f(x) = x|x|\) and find that which of the following(s) are/is correct :
(A) \(f(x) = x|x|\) is differentiable at x = 0.
(B) \(f(x) = x|x|\) is continuous at x = 0 but not differentiable at x = 0.
(C) \(f(x) = x|x|\) has point of inflection at x = 0.
(D) \(f(x) = x|x|\) is symmetrical about y-axis.
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
Removing the modulus: \(f(x) = x^{2}\) for \(x \ge 0\) and \(f(x) = -x^{2}\) for x < 0.
Derivative at x = 0 from the definition:
\( \displaystyle f'(0) = \lim_{h \to 0}\frac{h|h| - 0}{h} = \lim_{h \to 0}|h| = 0 \)
Second derivative on each side:
\(f''(x) = 2\) for x > 0
\(f''(x) = -2\) for x < 0
So f is differentiable at 0 with f′(0) = 0, the curve changes from concave down to concave up at x = 0, and f(−x) = −f(x).
Checking the statements
(A) \(f(x) = x|x|\) is differentiable at x = 0. — Correct. The limit of |h| is 0, so f′(0) = 0.
(B) \(f(x) = x|x|\) is continuous at x = 0 but not differentiable at x = 0. — Incorrect. f′(0) = 0 exists.
(C) \(f(x) = x|x|\) has point of inflection at x = 0. — Correct. The concavity changes at x = 0 (f′′ is −2 on the left, 2 on the right).
(D) \(f(x) = x|x|\) is symmetrical about y-axis. — Incorrect. f(−1) = −1 but f(1) = 1; the graph is symmetric about the origin.