Question

f(x) = (x^2 - 1) ^3

a) domain
b) intercepts
c) asymptotes
d) symmetry
e) critical values of the 1st derivative
f) intervals of increase/decrease
g) local and extreme maximum/minimum points
h) critical values of the 2nd derivative
i) concavity
j) points of inflection

Answer 1

I got the derivative for e but not sure how to pull out the critical values from 6x^5 -12x^3 + 6x and then the intervals, locals and extremes always confuses me

Answer 2

(a) domain: all reals (as with any polynomial)
(b) x = ±1
(c) no asymptotes (as with any polynomial)
(d) y-axis
(e) f' = 6x(x^2-1)^2, so f'=0 at x = -1, 0, 1
(f) f' > 0 for x > 0 (except at x=1, where f' = 0)
f' < 0 for x < 0 (except at x=-1, where f' = 0)
(g) f(0) = -1
(h) f" = 6(x^2-1)(5x^2-1) so f"=0 at x = ±1, ±1/√5
(i) f" > 0 (concave up) for x < -1, -1/√5 < x < 1/√5, x > 1
down elsewhere
(j) where f"=0