In the rectangular coordinate plane, rectangles are drawn such that two vertices lie on the x-axis and the other two vertices lie on the parabola $y = 27 - x ^ { 2 }$, and the rectangles lie between this parabola and the x-axis.
Accordingly, what is the perimeter of the rectangle with the largest area?
A) 40
B) 42
C) 44
D) 46
E) 48

A crystal in the shape of a cube with one edge of length $x$ units has a production cost of 5 TL per unit cube based on volume, and a selling price of 20 TL per unit square based on surface area.
Accordingly, for what value of x in units will the profit from selling this crystal be maximum?\ A) 16\ B) 18\ C) 20\ D) 22\ E) 24

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) $\mathrm { f } ( 2 ) < \mathrm { f } ( 1 ) < \mathrm { f } ( 0 )$

A polynomial $P(x)$ with real coefficients and of degree four satisfies the inequality
$$P(x) \geq x$$
for every real number $x$.
$$\begin{aligned} & P(1) = 1 \\ & P(2) = 4 \\ & P(3) = 3 \end{aligned}$$
according to, $\mathbf{P(4)}$ is equal to what?

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) $-6$