We see that f ′ and f ′′ are both positive only if x > 1.
(E) Note from the sign lines in Question 63 that f changes from
decreasing to increasing at x = 1, so f has a local minimum there.
Also, the graph of f changes from concave up to concave down at x = 0,
then back to concave up at ; hence f has two points of inflection.
(C) The derivatives of ln (x + 1) are , . . .
The nth derivative at .
(C) The absolute-value function f (x) = |x| is continuous at x = 0, but f ′(0)
does not exist.
(C) Let f ′(x) = f (x); then F ′(x + k) = f (x + k);
Or let u = x + k. Then dx = du; when x = 0, u = k; when x = 3, u = 3 + k.
(E) See the figure. The equation of the generating circle is (x − 3)^2 + y 2
= 1, which yields .
