A) -2
B) -1
C) 0
D) ½
Correct Answer: D
Solution :
[d] For Rolle?s theorem in [a, b], f(a)=f(b), In \[[0,1]\Rightarrow f(0)=f(1)=0\] \[\because \] the function has to be continuous in [0, 1] \[\Rightarrow f(0)=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f(x)=0\Rightarrow \underset{x\to 0}{\mathop{\lim }}\,{{x}^{\alpha }}\log x=0\] \[\Rightarrow \underset{x\to 0}{\mathop{\lim }}\,\frac{\log x}{{{x}^{-\alpha }}}=0\] Applying L.H. Rule \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1/x}{-\alpha {{x}^{-\alpha -1}}}=0\] \[\Rightarrow \underset{x\to 0}{\mathop{\lim }}\,\frac{-{{x}^{\alpha }}}{\alpha }=0\Rightarrow \alpha >0\]
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