A) \[\pm \,\,1\]
B) \[\pm \,\,\sqrt{2}\]
C) \[0\] and \[1\]
D) \[\pm \,\,2\]
Correct Answer: B
Solution :
Given, \[f(x)=\int_{1}^{x}{\sqrt{4-{{t}^{2}}}}\,\,dt\] On differentiating w. r. t. x, we get \[f'(x)=\sqrt{4-{{x}^{2}}}\] (1) \[\therefore \] \[x-f'(x)=x-\sqrt{4-{{x}^{2}}}=0\] \[\Rightarrow \] \[x=\sqrt{4-{{x}^{2}}}\] \[\Rightarrow \] \[{{x}^{2}}=4-{{x}^{2}}\] \[\Rightarrow \] \[2{{x}^{2}}=4\] \[\Rightarrow \] \[{{x}^{2}}=2\] \[\Rightarrow \] \[x=\pm 2\] Hence, real roots of \[\{x-f'(x)\}\] and \[\pm \sqrt{2}\].
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