43.44.45.(C) Counterexamples are, respectively, for (A), f(x) = x^3 , c = 0; for (B),
f(x) = x^4 , c = 0; for (D), f(x) = x^2 on (−1, 1).(D) f ′(x) > 0; the curve shows that f ′ is defined for all a < x < b, so f is
differentiable and therefore continuous.(C) Since g′(x) < 0 for all x, the table must have decreasing values as x
increases, and , so there must be at least one value above the
x-axis and one value below the x-axis.