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Functions

Which of the ...

Question

Which of the following function defined below are NOT differentiable at the indicated point ?

A

f (x) = {\begin{matrix} x^{2}; - 1 \leq x < 0 - x^{2}; 0 \leq x \leq 1 \end{matrix}

at

x = 0

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B

g (x) = {\begin{matrix} x; - 1 \leq x < 0 tan x; 0 \leq x \leq \frac{π}{4} \end{matrix}

at

x = 0

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C

h (x) = {\begin{matrix} sin 2 x; x \leq 0 2 x; x > 0 \end{matrix}

at

x = 0

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D

k (x) = {\begin{matrix} x; 0 \leq x \leq 1 2 - x; 1 < x \leq 2 \end{matrix}

at

x = 1

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Solution

The correct option is C $k (x) = {\begin{matrix} x; 0 \leq x \leq 1 2 - x; 1 < x \leq 2 \end{matrix}$ at $x = 1$
$f^{'} (x) = {\begin{matrix} 2 x; - 1 \leq x < 0 - 2 x; 0 \leq x \leq 1 \end{matrix}$ at $x = 0$
$L H S = f^{'} (0) = 0 = R H S$
.
$g^{'} (x) = {\begin{matrix} 1; - 1 \leq x < 0 {sec}^{2} x; 0 \leq x \leq \frac{π}{4} \end{matrix}$ at $x = 0$
$L H S = g^{'} (0) = 1 = R H S$
.
$h^{'} (x) = {\begin{matrix} 2 cos 2 x; x \leq 0 2; x > 0 \end{matrix}$ at $x = 0$
$L H S = h^{'} (0) = 2 = R H S$
.
$k^{'} (x) = {\begin{matrix} 1; 0 \leq x \leq 1 - 1; 1 < x \leq 2 \end{matrix}$ at $x = 1$
$L H S = 1 R H S = - 1$

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