What are the local extrema of #f(x)= x^3-3x^2-9x+7#?

1 Answer
Feb 11, 2017

Answer:

Local maximum : #f(-1)=12#. Local minimum :# f(3)=-20#.

Explanation:

#f = x^3(1-3/x-9/x^2+7/x^3) to +-oo#, as #x to +-oo#.

f'=3(x^2-2x-3)=0, at x = -1 and 3.

#f''=6x-6, <9#, at #x = -1, >0#, at #x = 3 and = 0#, at #x =1.#

So, #local-max f = f(-1)=12 and local-min f = f(3)=-20#.

As, #f''' ne 0, ( 1, -4 )# is a POI ( point of inflexion ).

graph{(x^3-3x^2-9x+7-y)((x-1)^2+(y+4)^2-.01)=0 [-34, 34, -21, 13]}