A twice-differentiable function f on the interval (−4,4) satisfies the following conditions:
• f is decreasing on (−4,−1), increasing on (−1,2), and decreasing on (2,4).
• f changes concavity from concave down to concave up at x=0.
Which of the following sign charts for f′ and f′′ is most consistent with this information?
a) f′ is positive on (−4,−1), 0 at x=−1, negative on (−1,2), 0 at x=2, and positive on (2,4); f′′ is negative on (−4,0), 0 at x=0, and positive on (0,4).
b) f′ is negative on (−4,−1), 0 at x=−1, positive on (−1,2), 0 at x=2, and negative on (2,4); f′′ is negative on (−4,0), 0 at x=0, and negative on (0,4).
c) f′ is negative on (−4,−1), 0 at x=−1, positive on (−1,2), 0 at x=2, and negative on (2,4); f′′ is negative on (−4,0), 0 at x=0, and positive on (0,4).
d) f′ is negative on (−4,−1), 0 at x=−1, positive on (−1,2), 0 at x=2, and negative on (2,4); f′′ is positive on (−4,0), 0 at x=0, and negative on (0,4).