Let $a$ and $b$ be real numbers. It is known that the polynomial
$$f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 1$$
is

• increasing on the interval $( - \infty , 1 )$,
• decreasing on the interval $( 1,5 )$,
• increasing on the interval $( 5 , \infty )$.

Accordingly, what is $f ( 2 )$?
A) 0
B) 3
C) 6
D) 9
E) 12

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = x^{3} + 9x^{2} + ax + b$$
takes positive values on positive real numbers and negative values on negative real numbers.
What is the smallest integer value that $a$ can take?
A) 9 B) 13 C) 17 D) 21 E) 25

Let $a$ and $b$ be real numbers. The function $f$ defined as
$$f(x) = ax^{3} + bx^{2} + x + 7$$
is always increasing.
If $f(-1) = 0$, what is the sum of the different integer values that $b$ can take?
A) 11 B) 13 C) 15 D) 17 E) 19

Let $k$ and $m$ be real numbers. The functions $f$ and $g$ defined on the set of real numbers are
$$\begin{aligned} & f(x) = 2x^{3} - 9x^{2} - mx - k \\ & g(x) = x^{3} \cdot f(x) \end{aligned}$$
The functions $f$ and $g$ have local extrema at $x = -1$.
What is the sum $k + m$?
A) 31 B) 33 C) 35 D) 37 E) 39

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) $-6$