Several false statements were corrected, such as the misconception that a zero second derivative at a point confirms an inflection point and that the concavity of the first derivative determines the sign of the second derivative. Also, an inflection point does not guarantee that the derivative has a local extremum.
(a) False: If f ′′(2)=0, x=2 is not necessarily an inflection point of f. An inflection point requires f ′′ to change sign.
(b) False: If f ′′ is concave up on an interval I, this describes the concavity of f ′ and does not imply f ′′ is positive on I.
(c) True: If f is concave up on an interval I, then f ′′ is positive on I because the slope of the tangent lines is increasing.
(d) True: If f ′′(2) does not exist and x=2 is in the domain of f, x=2 can be a critical point of f′ if f′ is also undefined or zero at x=2.
(e) False: If f has an inflection point at x=3 and f is differentiable at x=3, f′ does not necessarily have a local minimum or maximum at x=3.
(f) True: If f ′(1)=0 and f ′′(1)=-2, f has a local maximum at x=1 because the second derivative is negative, indicating concave down.
(g) False: The second-derivative test involves checking the sign of f′′ at the critical points, not on each side of them.
(h) False: The second-derivative test does not always produce the same information as the first-derivative test; it can be inconclusive when f′′=0.