A) f is derivable at x=0
B) f is continuous but not derivable at x=0
C) LHD at x=0 is 1
D) RHD at x=0 is 1
Correct Answer: A
Solution :
[a] We have \[f(x)=\left\{ \begin{matrix} {{x}^{3}},\,\,\,\,\,x>0 \\ 0,\,\,\,\,\,\,\,x=0 \\ -{{x}^{2}},\,\,\,x<0 \\ \end{matrix} \right.\] Clearly, f(x) is continuous at x=0. (L.H.D. at x=0)=0\[{{\left[ \frac{d}{dx}(-{{x}^{3}}) \right]}_{x=\,0}}={{[-\,3{{x}^{2}}]}_{x=\,0}}=0\] Similarly, (R.H.D. at x=0) =0. So, f(x) is differentiable at x=0.
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