APFIVE Concavity and Mean Value Theorem
Let $$t$$ be a twice-differentiable function on $$[-3, 3]$$ with $$t(-3) = 12$$, $$t(0) = 9$$, and $$t(3) = 18$$. If $$t''(x) < 0$$ for $$x \in (-3, 0)$$ and $$t''(x) > 0$$ for $$x \in (0, 3)$$, which of the following statements about $$t'(0)$$ must be true?
A
The sign of $$t'(0)$$ cannot be determined from the given information
B
$$t'(0) = 0$$
C
$$t'(0) > 0$$
D
$$t'(0) < 0$$
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