Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure.
Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value.
Which of the following statements are definitely true?
A) Only II
B) Only III
C) I and II
D) II and III
E) I, II and III

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given in the following Cartesian coordinate plane.
Accordingly; what is the correct ordering of the values $\mathbf { f } ( \mathbf { 0 } )$, $\mathbf { f } ( \mathbf { 1 } )$ and $\mathbf { f } ( \mathbf { 2 } )$?
A) $\mathrm { f } ( 0 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 2 )$ B) $\mathrm { f } ( 0 ) < \mathrm { f } ( 2 ) < \mathrm { f } ( 1 )$ C) $f ( 1 ) < f ( 2 ) < f ( 0 )$ D) $\mathrm { f } ( 2 ) < \mathrm { f } ( 0 ) < \mathrm { f } ( 1 )$ E) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$

In the rectangular coordinate plane, the graph of the derivative $f'$ of a continuous function $f$ defined on the set of real numbers is shown in the figure.
$$f(5) = f(20) = 0$$
Given that, what is the local minimum value of the function $f$?
A) $-18$ B) $-15$ C) $-12$ D) $-9$ E) $-6$