Question
Check whether the function given by:
has local maxima and local minima.
Answer :
Word Count : 126
Differentiate: $f(x)=(x-4)^3(x+1)^2$. By product rule $$ f'(x)= (x-4)^3\cdot 2(x+1)+3(x-4)^2\cdot (x+1)^2 $$ Factor common terms $(x-4)^2(x+1)$: $$ f'(x)=(x-4)^2(x+1)\big(2(x-4)+3(x+1)\big)=5(x-4)^2(x-1)(x+1). $$ Critical points: $x=-1,\,1,\,4$. Note $(x-4)^2$ is nonnegative ________ _______ ___ ______ ________ _____ _________ __________ __________ ____.
_____ ______ __________ _____ _____ ____ _________ __________ ___ ______ _________.
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_______ ________ ______ __________ ________.
_______ _______ _______ ________ __________ __________ __________ _________ _________ ___ ____.
_______ ___ __________ __________ ________ ___ ______ ______ _____ ________ __________.
____ __________ _________ ______ ______ ____ ____ ______.
_________ _____ _________ ___ _______ _______ __________ ____ ________.
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__________ ____ ________ ________ _______ ________ ________ ______ _______ _______ __________.
___ ______ ____ _______ _____ ________.
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Differentiate: $f(x)=(x-4)^3(x+1)^2$. By product rule $$ f'(x)= (x-4)^3\cdot 2(x+1)+3(x-4)^2\cdot (x+1)^2 $$ Factor common terms $(x-4)^2(x+1)$: $$ f'(x)=(x-4)^2(x+1)\big(2(x-4)+3(x+1)\big)=5(x-4)^2(x-1)(x+1). $$ Critical points: $x=-1,\,1,\,4$. Note $(x-4)^2$ is nonnegative ________ _______ ___ ______ ________ _____ _________ __________ __________ ____.
_____ ______ __________ _____ _____ ____ _________ __________ ___ ______ _________.
______ __________ ___ ______ _____ _____ _________ ____.
_______ ________ ______ __________ ________.
_______ _______ _______ ________ __________ __________ __________ _________ _________ ___ ____.
_______ ___ __________ __________ ________ ___ ______ ______ _____ ________ __________.
____ __________ _________ ______ ______ ____ ____ ______.
_________ _____ _________ ___ _______ _______ __________ ____ ________.
__________ ____ ____ _________ ____ ___ _____ __________ ______ ___ ________.
__________ ____ ________ ________ _______ ________ ________ ______ _______ _______ __________.
___ ______ ____ _______ _____ ________.
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Examine the function f : R→R defined by
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