What are the critical points of #f(x) = x(x + 1)^3#?

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Answer 1 · 2018-01-23T15:27:12

The critical points are #=(-1,0)# and #=(-1/4, -27/256)#

Calculate the first derivative and determine the critical points #f'(x)=0#

Here,

#f(x)=x(x+1)^3#

Apply the product rule for differentiation

#(uv)'=u'v+uv'#

#u=x#, #=>#, #u'=1#

#v=(x+1)^3#, #=>#, #v'=3(x+1)^2#

#f'(x)=1*(x+1)^3+3x(x+1)^2#

#=(x+1)^2(x+1+3x)#

#=(x+1)^2(4x+1)#

#f'(x)=0#, #=>#

The critical points are when #x=-1# and #x=-1/4#

graph{x(x+1)^3 [-1.994, 1.044, -0.595, 0.924]}