First slide

Differentiability

Question

Which of the following function is thrice differentiable at x = 0?

Moderate

A

$f (x) = |x^{3}|$
B

$f (x) = x^{3} |x|$
C

$f (x) = |x| \sin^{3} x$
D

$f (x) = x |\tan^{3} x|$

Solution

$\begin{array}{l} f (x) = |x^{3}| = \{\begin{cases} - x^{3}, x < 0 \\ x^{3}, x \geq 0 \end{cases} or f''' (x) = \{\begin{cases} - 6, x < 0 \\ 6, x > 0 \end{cases} \\ Hence, f'' (0) does not exist . \\ f (x) = x^{3} |x| = \{\begin{cases} - x^{4}, x < 0 \\ x^{4}, x \geq 0 \end{cases} or f''' (x) = \{\begin{cases} - 24 x, x < 0 \\ 24 x, x > 0 \end{cases} . \\ Hence, f''' (0) = 0 and it exists . \\ Similarly, for f (x) = |x| \sin^{3} x and f (x) = x |\tan^{3} x| also, f''' (0) = 0 \\ and it exists \end{array}$

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