First slide

Differentiability

Question

$The function f (x) = \sqrt{1 - \sqrt{1 - x^{2}}} is$

Moderate

A

$Continuous on [- 1, 1] and differentiable on (- 1, 1)$
B

$Continuous on [- 1, 1] and differentiable on (- 1, 0) \cup (0, 1)$
C

$Continuous as well as differentiable on [- 1, 1]$
D

None of these

Solution

$\begin{array}{l} f (x) = \sqrt{1 - \sqrt{1 - x^{2}}} \\ Clearly, the domain of definition is [- 1, 1] . \\ f' (x) = \frac{1}{2 \sqrt{1 - \sqrt{1 - x^{2}}}} (- \frac{1}{2 \sqrt{1 - x^{2}}}) (- 2 x) \\ = \frac{x}{\sqrt{1 - x^{2}} \sqrt{1 - \sqrt{1 - x^{2}}}}, \\ Which does not exist for x= 0, - 1, 1 \\ Also, f(x) is continuous on [- 1, 1] \\ So, (b) is the correct choice . \end{array}$

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