#x^3 > 2x^2 +x or x^3 - 2x^2 - x > 0 or x( x^2-2x-1) >0 # or
Roots of # (x^2-2x-1) # are # x = (2 +- sqrt((2^2-4*1*-1)))/2#
or #x = 1+- sqrt2 or x = 2.4142 , x= - 0.4142#
#:. x( x^2-2x-1) >0 or x( x-2.4142)(x+0.4142) >0#
Critical points are #x=0 , x=2.4142 , x= -0.4142#
Sign chart:
When # x < -0.4142# sign of #x( x-2.4142)(x+0.4142) # is
# (-) * (-) * (-) = (-) ; <0 #
When # -0.4142 < x <0 # sign of #x( x-2.4142)(x+0.4142) # is
# (-) * (-) * (+) = (+) ; >0 #
When # 0 < x <2.4142 # sign of #x( x-2.4142)(x+0.4142) # is
# (+) * (-) * (+) = (-) ; <0 #
When # x >2.4142 # sign of #x( x-2.4142)(x+0.4142) # is
# (+) * (+) * (+) = (+) ; >0 #
Solution: # -0.4142 < x <0 and # x >2.4142 #
In interval notation: #(-0.4142,0) uu (2.4142,oo)#
graph{x^3-2x^2-x [-10, 10, -5, 5]} [Ans]
