CUET · MATHS · PYQ PAPER 2023
Let \(f(x)=x|x|\). Then \(f(x)\) is:
- A Continuous at \(x=0\) but not differentiable at \(x=0\)
- B Differentiable at \(x=0\) but not continuous at \(x=0\)
- C Both continuous and differentiable at \(x=0\)
- D Neither continuous at \(x=0\) nor differentiable at \(x=0\)
Answer & Solution
Correct Answer
(C) Both continuous and differentiable at \(x=0\)
Step-by-step Solution
Detailed explanation
\(f(x) = x|x| = \begin{cases} x^2 & \text{if } x \ge 0 \\ -x^2 & \text{if } x Continuity at \(x=0\): \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x^2) = 0\) \(\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2) = 0\) \(f(0) = 0|0| = 0\). Continuous. Differentiability at…
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