How do you find the exact relative maximum and minimum of the polynomial function of #g(x) = x^3 - 3x^2 - 9x +1#?

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Answer 1 · 2016-03-16T17:24:09

#P_("max") ->(x,y) ->(-1,6)#
#P_("min")->(x,y)->(3,-26)#

#(dy)/(dx)=3x^2-6x-9 = 0#

Divide throughout by 3 giving

#x^2-2x-3=0#

#(x+1)(x-3)=0#

#x=-1" or "+3#
'~~~~~~~~~~~~~~~~~~~~~~
#(d^2y)/(dx^2)=6x-6#

At #x=-1 ; (d^2y)/(dx^2)<0 => "maximum"#

At #x=+3 ; (d^2y)/(dx^2)>0 => "minimum"#

By substitution you find

#P_("max") ->(x,y) ->(-1,6)#
#P_("min")->(x,y)->(3,-26)#

Tony B