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Question

lf $$\mathrm{f}({x})=\sqrt{1-\sqrt{1-{x}^{2}}}$$, then $$\mathrm{f}({x})$$ is


A
Continuous on [1,1] and differentiable on (1,1)
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B
Continuous on [1,1] and differentiable on (1,0)(0,1)
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C
Continuous and differentiable on [1,1]
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D
Continuous and differentiable on (1,1)
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Solution

The correct option is A Continuous on $$[-1,1]$$ and differentiable on $$(-1,1)$$
$$f(x)\quad =\sqrt { 1-\sqrt { 1-{ x }^{ 2 } }  } \quad for\quad x\in (-1,1)\\ let\quad x\quad =\quad sin(t)\quad for\quad t\quad \in \quad (\frac { -\pi  }{ 2 } ,\frac { \pi  }{ 2 } )\\ f(t)\quad =\quad \sqrt { 1-cos(t) } =\quad \sqrt { 2 } sin(t/2)\quad \\ The\quad above\quad function\quad is\quad continuous\quad as\quad well\quad as\quad differentiable\quad for\quad x\quad \in \quad (-1,1)$$
Hence, the answer is "A"

Mathematics

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