Question

Suppose that f(x), f'(x), and f''(x) are continuous for all real numbers x, and the f has the following properties.

I. f is negative on (-inf, 6) and positive on (6,inf).
II. f is increasing on (-inf, 8) and positive on 8, inf).
III. f is concave down on (-inf, 10) and concave up on (10, inf).

Of the following, which has the smallest numerical value?

(a) f'(0)
(b) f'(6)
(c) f''(4)
(d) f''(10)
(e) f''(12)

Answer 1

i) --> f'(0) > 0
i) --> f'(6) > 0
iii) --> f"(4) < 0
iii) --> f"(10) = 0
iii) --> f"(12) > 0

so f"(4) is the only negative value

Answer 2

sorry, (ii) relates to f', not (i)