First slide

Differentiability

Question

$The domain of the derivative of the function f (x) = \{\begin{cases} \tan^{- 1} x if | x | \leq 1 \\ \frac{1}{2} (| x | - 1) if | x | > 1 \end{cases}$

Moderate

A

$R - \{0\}$
B

$R - \{1\}$
C

$R - \{- 1\}$
D

$R - \{1, - 1\}$

Solution

$\begin{array}{l} f (x) = \frac{1}{1 + x^{2}}, i f |x| < 1 a n d i s w i l l d e f i n e d . \\ f (x) = \frac{1}{2}, i f x > 1 \\ = - \frac{1}{2}, i f x < - 1. \\ R i g h t d e r i v a t i v e a t x = 1 i s_{x \to 1}^{\lim} \frac{1}{2} = \frac{1}{2} \\ L e f t d e r i v a t i v e a t x = 1 i s_{x \to 1}^{\lim} \frac{1}{1 + x^{2}} = \frac{1}{2} \\ H e n c e, f (x) a t x = 1 i s \frac{1}{2} . \\ f (x) a t x = - 1 d o e s n o t e x i s t \\ ∴ f (x) i s w e l l d e f i n e d a t a l l p o i n t s e x c e p t - 1. \\ H e n c e (3) i s t h e c o r r e c t a n s w e r . \end{array}$

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