Find the intervals in which f(x) is increasing or decreasing: f(x)=xlxl, x E R.

Asked by raginee311 | 31st Aug, 2015, 08:26: AM

Expert Answer:

C o n s i d e r space t h e space g i v e n space f u n c t i o n f open parentheses x close parentheses equals x open vertical bar x close vertical bar comma space w h e r e space x element of R R e w r i t i n g space t h e space a b o v e space f u n c t i o n comma space w e space h a v e comma f open parentheses x close parentheses equals x open parentheses x squared close parentheses to the power of 1 half end exponent D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma f apostrophe open parentheses x close parentheses equals square root of x squared end root cross times 1 plus x cross times 1 half cross times open parentheses x squared close parentheses to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent cross times open parentheses 2 x close parentheses space space space space space space space space equals square root of x squared end root plus fraction numerator 2 x squared over denominator 2 square root of x squared end root end fraction space space space space space space space space equals square root of x squared end root plus fraction numerator x squared over denominator square root of x squared end root end fraction space space space space space space space space equals fraction numerator 2 x squared over denominator square root of x squared end root end fraction space space space space space space space space equals fraction numerator 2 x squared over denominator open vertical bar x close vertical bar end fraction space space space space space space space space equals 2 open vertical bar x close vertical bar T h u s comma space f apostrophe open parentheses x close parentheses less or equal than 0 comma space i f space x less-than or slanted equal to 0 space a n d space f apostrophe open parentheses x close parentheses greater or equal than 0 comma space i f space x greater or equal than 0 T h e r e f o r e comma space f open parentheses x close parentheses space i s space d e c r e a sin g space i n space t h e space i n t e r v a l space left parenthesis negative infinity comma 0 right square bracket space space a n d space i t space i s space i n c r e a sin g space i n space t h e space i n t e r v a l space left square bracket 0 comma infinity right parenthesis.

Answered by Vimala Ramamurthy | 31st Aug, 2015, 09:34: AM