AP EAMCET · Maths · Continuity and Differentiability
If \(f(x)=\left\{\begin{array}{cc}x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.\); Which of the following is true?
- A \(f(x)\) is continuous and differentiable if \(0 \leq \alpha \lt 1\)
- B \(f(x)\) is discontinuous and not differentiable if \(0 \leq \alpha \lt 1\)
- C \(f(x)\) is continuous and differentiable for \(\alpha\gt1\)
- D \(f(x)\) is discontinuous and differentiable for \(\alpha\gt1\)
Answer & Solution
Correct Answer
(C) \(f(x)\) is continuous and differentiable for \(\alpha\gt1\)
Step-by-step Solution
Detailed explanation
Given, \(f(x)=\left\{\begin{array}{cc}x^\alpha \sin \left(\frac{1}{x}\right) ; & x \neq 0 \\ 0 ; & x=0\end{array}\right.\) \(\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} x^\alpha \sin \left(\frac{1}{x}\right)=0 \text { if } \alpha\gt0\) So, \(f(x)\) is continuous for…
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