We are to find the range of the values of $k$ such that the inequality
$$\frac { \log 3 x } { 4 x + 1 } \leqq \log \left( \frac { 2 k x } { 4 x + 1 } \right) \tag{1}$$
holds for all positive real numbers $x$, where $\log$ is the natural logarithm.
(1) For $\mathbf{A}$ and $\mathbf{B}$ in the following sentences, choose the correct answer from among (0) $\sim$ (8) below.
By transforming inequality (1) we obtain
$$\log k \geqq \mathbf { A } . \tag{2}$$
Here, when the right side of (2) is denoted by $g ( x )$ and this $g ( x )$ is differentiated with respect to $x$, we have
$$g ^ { \prime } ( x ) = \mathbf { B } .$$
(0) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) - \log 2 x$
(1) $\frac { \log 3 x } { 4 x + 1 } - \log ( 4 x + 1 ) + \log 2 x$
(2) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) + \log 2 x$
(3) $\frac { \log 3 x } { 4 x + 1 } + \log ( 4 x + 1 ) - \log 2 x$
(4) $\frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$
(5) $\frac { 3 x + 2 + \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (6) $- \frac { 4 \log 3 x } { ( 4 x + 1 ) ^ { 2 } }$ (7) $\frac { 3 x - 2 - \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$ (8) $- \frac { 3 \log 2 x } { ( 4 x + 1 ) ^ { 2 } }$
(2) In the following sentences, for $\mathbf { E } , \mathbf { F }$ and $\mathbf { G }$, choose the correct answer from among (0) $\sim$ (3) below. For the other blanks, enter the correct number.
Over the interval $0 < x < \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { E }$ and over the interval $\frac { \mathbf{C} } { \mathbf{D} } < x$, $g ( x )$ is $\mathbf { F }$. Hence at $x = \frac { \mathbf { C } } { \mathbf{D} }$, $g ( x )$ is $\mathbf { G }$.
From the above, the range of the value of $k$ such that inequality (1) holds for all positive real numbers $x$ is
$$k \geqq \frac { \mathbf { H } } { \mathbf { I } }$$
(0) increasing
(1) decreasing
(2) maximized
(3) minimized

Find the derivative $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$ of the following real function $y(x)$ defined for $0 < x < 1$: $$y(x) = (\arccos x)^{\log x}$$ where $0 < \arccos x < \pi/2$.

$$\lim_{x \rightarrow 1} \frac{1-\sqrt{x}}{\ln x}$$
What is the value of this limit?
A) $\frac{-1}{2}$
B) $0$
C) $\frac{1}{2}$
D) $1$
E) $2$

$$f(x) = \ln\left(\sin^{2} x + e^{2x}\right)$$
Given this, what is $f'(0)$?
A) $e$
B) $1$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $2$

$$f ( x ) = e ^ { 2 x } - e ^ { - 2 x }$$
What is the value of the 15th order derivative of the function at the point $x = \ln 2$, that is $\mathbf { f } ^ { \mathbf { ( 1 5 ) } } ( \mathbf { \ln } \mathbf { 2 } )$?
A) $17 \cdot 2 ^ { 13 }$
B) $15 \cdot 2 ^ { 13 }$
C) $9 \cdot 2 ^ { 13 }$
D) $15 \cdot 2 ^ { 12 }$
E) $7 \cdot 2 ^ { 12 }$

$$\lim _ { x \rightarrow \infty } \frac { \ln ( x - 3 ) } { \ln \sqrt { x } }$$
What is the value of this limit?
A) 1
B) 2
C) 3
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 2 }$

What is the value of the 16th order derivative $f ^ { ( 16 ) } ( x )$ of the function $f ( x ) = e ^ { x } \cdot \cos x$ at the point $x = 0$?
A) 32 B) 64 C) 128 D) 256 E) 512

Let $a$ and $b$ be real numbers. A function $f$ is defined on the set of positive real numbers as
$$f ( x ) = a x ^ { a } + b x ^ { b }$$
$$\begin{aligned} & f ( 1 ) = 6 \\ & f ^ { \prime } ( 1 ) = 20 \end{aligned}$$
Given that, what is $f''(1)$?
A) 44
B) 46
C) 48
D) 50
E) 52