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Asked by g_archanasharma | 09 Apr, 2019, 21:10: PM
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G i v e n space f left parenthesis x right parenthesis equals vertical line x vertical line i. e. space f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell x space space space space space space i f space x greater or equal than 0 end cell row cell negative x space space space i f space x less than 0 end cell end table close N o w comma space w e space w i l l space f i n d space t h e space L H L space a n d space R H L thin space t o space c h e c k space t h e space c o n t i n u i t y space o f space f left parenthesis x right parenthesis space a t space x equals 0 L H L equals limit as x rightwards arrow 0 to the power of minus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of left parenthesis negative x right parenthesis equals 0 R H L equals limit as x rightwards arrow 0 plus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of left parenthesis x right parenthesis equals 0 equals f left parenthesis 0 right parenthesis rightwards double arrow L H L equals R H L equals f left parenthesis 0 right parenthesis T h u s comma space f left parenthesis x right parenthesis space i s space c o n t i n u o u s space a t space x equals 0 space...... space left parenthesis i right parenthesis A l s o comma space g left parenthesis x right parenthesis equals vertical line x cubed vertical line i. e. space g left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell x cubed space space space space space space i f space x greater or equal than 0 end cell row cell negative x cubed space space space i f space x less than 0 end cell end table close N o w comma space w e space w i l l space f i n d space t h e space L H L space a n d space R H L thin space t o space c h e c k space t h e space c o n t i n u i t y space o f space g left parenthesis x right parenthesis space a t space x equals 0 L H L equals limit as x rightwards arrow 0 to the power of minus of g left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of left parenthesis negative x cubed right parenthesis equals 0 R H L equals limit as x rightwards arrow 0 plus of g left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of left parenthesis x cubed right parenthesis equals 0 equals g left parenthesis 0 right parenthesis rightwards double arrow L H L equals R H L equals g left parenthesis 0 right parenthesis T h u s comma space g left parenthesis x right parenthesis space i s space c o n t i n u o u s space a t space x equals 0 space...... space left parenthesis i i right parenthesis N o w comma space t o space c h e c k space t h e space d i f f e r e n t i a b i l i t y space o f space f left parenthesis x right parenthesis space a n d space g left parenthesis x right parenthesis f apostrophe left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 1 space space space space space space i f space x greater than 0 end cell row cell negative 1 space space space i f space x less than 0 end cell end table close S o comma space f apostrophe left parenthesis x right parenthesis space h a s space t o space m a k e space a space j u m p space f r o m space minus 1 space t o space 1 space a t space x equals 0. A l s o comma space t h e space g r a p h space o f space f left parenthesis x right parenthesis space i s space n o t space s m o o t h space a t space x equals 0 T h u s comma space f left parenthesis x right parenthesis space i s space n o t space d i f f e r e n t i a b l e space a t space x equals 0. N o w comma g apostrophe left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 3 x squared space space space space space space i f space x greater than 0 end cell row cell negative 3 x squared space space space i f space x less than 0 end cell end table close N o w comma space t h e space g r a p h space o f space g left parenthesis x right parenthesis space w i l l space b e space a space s m o o t h space c u r v e space a n d space b o t h space t h e space p a r t s space i. e space f o r space x less than 0 space a n d space x greater or equal than 0 space w i l l space m e e t space a t space 0 space s m o o t h l y T h u s comma space g left parenthesis x right parenthesis space i s space d i f f e r e n t i a b l e space a t space x equals 0. H e n c e comma space o p t i o n space a space i s space c o r r e c t space i. e. space f left parenthesis x right parenthesis space a n d space g left parenthesis x right parenthesis space a r e space c o n t i n u o u s space a t space x equals 0.
Answered by Renu Varma | 10 Apr, 2019, 10:22: AM

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