Is #f(x)=(2x-2)(x+1)(x+4)# increasing or decreasing at #x=-1#?

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Answer 1 · 2017-04-05T05:27:41

#f(x) # is decreasing at #x=-1#

#f(x) = (2x-2)(x+1)(x+4)#

#= (2x-2)(x^2+5x+4)#

#f'(x) = (2x-2)(2x+5) + 2*(x^2+5x+4)# [Product rule]

#= 4x^2+6x-10 +2x^2+10x+8#

#= 6x^2+16x-2#

To test the slope of #f(x)# at #x=-1#

#f'(-1) = 6-16-2 = -12#

Hence the slope of #f(x)<0# at #x=-1# which implies that #f(x) # is decreasing at #x=-1#

This can be seen on the graph of #f(x)# below:
graph{(2x-2)(x+1)(x+4) [-5.665, 3.104, -1.95, 2.43]}