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Question: The set of all points where the function f(x) = x \|x\| is differentiable is...

The set of all points where the function f(x) = x |x| is

differentiable is

A

(–¥, ¥)

B

(–¥, 0) È (0, ¥)

C

(0, ¥)

D

[0, ¥)

Answer

(–¥, ¥)

Explanation

Solution

f(x) = x|x| = $\left{ \begin{matrix} x^{2} & ifx \geq 0 \

  • x^{2} & ifx < 0 \end{matrix} \right.\ $

Since x2 and –x2 are differentiable functions, f(x) is differentiable, except possibly at x = 0

Now f¢ (0+) = f(0+h)−f(0)h\frac{f(0 + h) - f(0)}{h}

= lim⁡h→0+\lim _ { h \rightarrow 0 ^ { + } } f(h)h\frac{f(h)}{h}

[Q f(0) = 0] = lim⁡h→0+\lim_{h \rightarrow 0^{+}} h2h\frac{h^{2}}{h}= lim⁡h→0+\lim_{h \rightarrow 0^{+}}h = 0

and f¢(0–) = lim⁡h→0−\lim _ { h \rightarrow 0 ^ { - } } f(0+h)−f(0)h\frac{f(0 + h) - f(0)}{h}

= lim⁡h→0−\lim _ { h \rightarrow 0 ^ { - } } f(h)−f(0)h\frac{f(h) - f(0)}{h}

= lim⁡h→0−\lim _ { h \rightarrow 0 ^ { - } } −h2h\frac{- h^{2}}{h}= lim⁡h→0–\lim_{h \rightarrow 0^{–}}h = 0

Hence f is differentiable everywhere.