Application of Derivatives

Question

Find the absolute maximum value and the absolute minimum value of
$straight f left parenthesis straight x right parenthesis space equals space open parentheses 1 half minus straight x close parentheses squared space space plus space straight x space cubed space space space in space space space left square bracket negative 2 comma space space 2.5 right square bracket.$

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space open parentheses 1 half minus straight x close parentheses squared space plus straight x space cubed$
The given function is differentiable for all x in [-2, 2.5]
$straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 space open parentheses 1 half minus straight x close parentheses space left parenthesis negative 1 right parenthesis space plus space 3 straight x space equals space minus 1 plus 2 straight x plus 3 straight x squared straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space space space space 3 straight x squared plus 2 straight x minus 1 space equals space 0 rightwards double arrow space space space space space space straight x space equals space fraction numerator negative 2 space plus-or-minus square root of 4 plus 12 end root over denominator 6 end fraction space equals space fraction numerator negative 2 plus-or-minus 4 over denominator 6 end fraction equals negative 1 comma space 1 third Both space space minus 1 comma space space space 1 third space space space element of space space space left square bracket negative 2 comma space 2.5 right square bracket$
$straight f left parenthesis negative 2 right parenthesis space equals space open parentheses 1 half plus 2 close parentheses squared plus left parenthesis negative 2 right parenthesis cubed space equals space 25 over 4 minus 8 space equals space minus 7 over 4 space equals space minus 1.75 straight f left parenthesis 2.5 right parenthesis space equals space open parentheses 1 half minus 2.5 close parentheses squared plus space left parenthesis 2.5 right parenthesis cubed space equals space 4 plus 125 over 8 space equals space 157 over 8 space equals space 19.625 straight f left parenthesis negative 1 right parenthesis space equals space open parentheses 1 half plus 1 close parentheses squared space minus 1 space equals space 9 over 4 minus 1 space equals space 5 over 4 straight f open parentheses 1 third close parentheses space equals space open parentheses 1 half minus 1 third close parentheses squared plus 1 over 27 space equals space 1 over 36 plus 1 over 27 space equals space fraction numerator 3 plus 4 over denominator 108 end fraction space equals space 7 over 108$
$therefore$ absolute maximum value = 19.625 and absolute minimum value = -1.75

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