Application of Derivatives

Question

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight x cubed left parenthesis straight x minus 1 right parenthesis squared$

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space straight x cubed space left parenthesis straight x minus 1 right parenthesis squared$
$therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x cubed straight d over dx left square bracket left parenthesis straight x minus 1 right parenthesis squared right square bracket space plus space left parenthesis straight x minus 1 right parenthesis squared. space straight d over dx left parenthesis straight x cubed right parenthesis$
$equals straight x cubed thin space 2 left parenthesis straight x minus 1 right parenthesis space plus space left parenthesis straight x minus 1 right parenthesis squared. space 3 straight x squared space equals space straight x squared left parenthesis straight x minus 1 right parenthesis space left square bracket 2 straight x plus 3 left parenthesis straight x minus 1 right parenthesis right square bracket space equals space straight x squared left parenthesis straight x minus 1 right parenthesis space left parenthesis 5 straight x minus 3 right parenthesis$
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When x<0 slightly, f '(x) = (+) (-) (-) = + ve
When x>0 slightly, f '(x) = (+) (-) (-) = + ve
$therefore$ at x = 0, f '(x) does not change sign
$therefore space space space space space straight x space equals space 0 space is space straight a space point space of space inflexion. space$
$When space straight x space less than space 1 space slightly comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis thin space left parenthesis plus right parenthesis space equals space minus ve When space straight x greater than 1 space slightly comma space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space equals space plus ve therefore space space space space space at space straight x space equals space 1 comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space plus ve therefore space space space space straight f left parenthesis straight x right parenthesis space has space local space minimum space value space at space straight x space equals space 1 space and space this space value space space equals space 0. When space straight x less than 3 over 5 space slightly comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis thin space left parenthesis negative right parenthesis space equals space plus ve When space straight x space greater than 3 over 5 space slightly comma space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis thin space left parenthesis plus right parenthesis space equals space minus ve$
$therefore space space space straight f left parenthesis straight x right parenthesis space has space local space maximum space value space at space straight x space equals space 3 over 5 and space this space value space space equals space open parentheses 3 over 5 close parentheses cubed space space open parentheses 3 over 5 minus 1 close parentheses squared space equals space 27 over 125 cross times 4 over 25 space equals space 108 over 3125.$

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Application Of Derivatives

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight x cubed left parenthesis straight x minus 1 right parenthesis squared$

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