How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for #f(x)=x^4-8x^3#?

1 Answer
tom · Stefan V.
Nov 15, 2016

points of inflection at #x=0# and #x=4#
concave up for #(-oo, 0)# and #(4, oo)#
concave down for #(0,4)#

Explanation:

You need the second derivative for concavity/inflection points

#f'(x)=4x^3- 24x^2#

#f^('')(x) = 12 x^2 - 48x#

#= 12 x(x-4)#

#f^('')(x)= 0# at #x =0# and #x=4#

Make a sign chart. To the left of #x=0#, the second derivative is positive. To the right of #x=4#, the second derivative is positive. Between #0# and #4# the second derivative is negative.

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