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Continuity And Differentiability

Question
CBSEENMA12035461
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If space space straight f left parenthesis straight x right parenthesis equals vertical line straight x vertical line cubed comma space show space that space straight f " left parenthesis straight x right parenthesis space exists space for space all space real space straight x space and space find space it.

Solution
We space have space straight f left parenthesis straight x right parenthesis equals open vertical bar straight x close vertical bar cubed equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell space space space straight x cubed end cell cell comma space straight x greater or equal than 0 end cell row cell negative straight x cubed end cell cell comma space straight x less than 0 end cell end table close space space space space space space space space space space space space space space space space space left square bracket because space open vertical bar straight x close vertical bar equals negative straight x space for space straight x less than 0 right square bracket because space straight f apostrophe left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell space space 3 space straight x cubed end cell cell comma space straight x greater or equal than 0 end cell row cell negative 3 straight x cubed end cell cell comma space straight x less than 0 end cell end table close space space space space space straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell space space 6 space straight x cubed end cell cell comma space straight x greater or equal than 0 end cell row cell negative 6 straight x cubed end cell cell comma space straight x less than 0 end cell end table close Hence space the space result

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