Homework Help: Calculus

Posted by Elaine on Tuesday, April 24, 2012 at 8:03pm.

assume that f(x)= x^3(x-2)^2
a) find the x-intercepts algebraically
b) find all the critical points
c) use the first derivative test to determine local minimums and maximums
d)find all inflection points
e)state the concavity of the graph on appropriate intervals

Calculus - Steve, Wednesday, April 25, 2012 at 10:53am
a) x=0, x=2

b)
f(x) = x^3(x-2)^2
f '(x) = 3x^2(x-2)^2 + 2x^3(x-2)
= x^2(x-2)(5x-6)

critical points at x=0,2,6/5

max/min are at (0,0)(2,0)(1.2,1.106)

inflection where f ''(x) = 0
f ''(x) = 4x(5x^2 - 12x + 6)
f ''(x) = 0 at x = .71, 1.69
plug in to obtain f(x) there

concave down: -oo < x < 0
concave up: 0 < x < .71
concave down: .71 < x < 1.69
concave up: 1.69 < x < oo

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