What are the critical values, if any, of #f(x)= -2 x^3 + 27 x^2 - 48 x + 7#?

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Answer 1 · 2015-12-23T20:51:43

#c=1,8#

A critical value is the #x#-value at which #f'(x)=0# or does not exist.

Find #f'(x)#.

#f'(x)=-6x^2+54x-48#

Find when #f'(x)=0#.

#-6x^2+54x-48=0#
#=>-6(x^2-9x+8)=0#
#=>-6(x-1)(x-8)=0#
#color(red)(x=1,8#

Find when #f'(x)# does not exist.

#f'(x)# is defined for all #x#.

Thus, the only critical values are #c=1,8#.

graph{-2 x^3 + 27 x^2 - 48 x + 7 [-5, 15, -289.3, 405]}