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
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