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Application Of Derivatives

Question
CBSEENMA12035392
Wired Faculty App

Find absolute maximum and minimum values of a function f given by
straight f left parenthesis straight x right parenthesis space equals space 12 straight x to the power of 4 over 3 end exponent space minus space 6 straight x to the power of 1 third end exponent comma space space space straight x space element of space space space open square brackets negative 1 comma space 1 close square brackets

Solution
straight f left parenthesis straight x right parenthesis space equals space 12 straight x to the power of 4 over 3 end exponent space minus space 6 straight x to the power of 1 third end exponent straight f apostrophe left parenthesis straight x right parenthesis space equals 12 space cross times 4 over 3 straight x to the power of 1 third end exponent space minus space 6 space cross times space 1 third straight x to the power of negative 2 over 3 end exponent space equals space 16 straight x to the power of 1 third end exponent minus 2 over straight x to the power of begin display style 2 over 3 end style end exponent space equals space fraction numerator 16 straight x minus 2 over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction space equals space fraction numerator 2 left parenthesis 8 straight x minus 1 right parenthesis over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction straight f apostrophe left parenthesis straight x right parenthesis space equals 0 space space space rightwards double arrow space fraction numerator 2 left parenthesis 8 straight x minus 1 right parenthesis over denominator straight x to the power of begin display style 2 over 3 end style end exponent end fraction space equals 0 space space space rightwards double arrow space space space space space 2 left parenthesis 8 straight x minus 1 right parenthesis space equals space 0 space space space space space rightwards double arrow space space space straight x space equals space 1 over 8 Also space straight f apostrophe left parenthesis straight x right parenthesis space is space not space defined space at space straight x space equals space 0
therefore space space space space space space critical space points space are space straight x space equals space 0 comma space space 1 over 8
Now,   Now comma space space space straight f left parenthesis negative 1 right parenthesis space equals space 12 left parenthesis negative 1 right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis negative 1 right parenthesis to the power of 1 third end exponent space equals space 12 plus 6 space equals space 18 space space space space space space space space space space space space space straight f left parenthesis 0 right parenthesis space equals space 0 minus 0 space equals space 0 space space space straight f open parentheses 1 over 8 close parentheses space equals space 12 open parentheses 1 over 8 close parentheses to the power of 4 over 3 end exponent space minus space 6 open parentheses 1 over 8 close parentheses to the power of 1 third end exponent space equals space 12 left parenthesis 2 to the power of negative 3 end exponent right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis 2 to the power of negative 3 end exponent right parenthesis to the power of 1 third end exponent space space space space space space space space space space space space space equals 12 left parenthesis 2 right parenthesis to the power of negative 4 end exponent space minus 6 space left parenthesis 2 right parenthesis to the power of negative 1 end exponent space equals space 12 space cross times space 1 over 16 minus 6 cross times 1 half equals space 3 over 4 minus 3 space equals space minus 9 over 4 space space space space space space space straight f left parenthesis 1 right parenthesis space equals space 12 left parenthesis 1 right parenthesis to the power of 4 over 3 end exponent space minus space 6 left parenthesis 1 right parenthesis to the power of 1 third end exponent space equals space 12 minus 6 space equals space 6

therefore space space space space absolute space maximum space value space space equals space 18 space space at space straight x space equals space minus 1 and space absolute space minimum space value space space equals space minus 9 over 4 space at space straight x space equals space 1 over 8.

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