APFIVE Properties of the First Derivative from Data
The table below gives values of a function $$ f(x) $$ at selected values of $$ x $$:
| $$ x $$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $$ f(x) $$ | 9 | 4 | 0 | -3 | -5 |
If $$ f $$ is twice-differentiable on the interval $$ 1 \leq x \leq 5 $$, which of the following statements could be true?
A
$$ f' $$ is positive and decreasing for $$ 1 \leq x \leq 5 $$
B
$$ f' $$ is positive and increasing for $$ 1 \leq x \leq 5 $$
C
$$ f' $$ is negative and decreasing for $$ 1 \leq x \leq 5 $$
D
$$ f' $$ is negative and increasing for $$ 1 \leq x \leq 5 $$
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