For positive numbers $a , b$ and natural numbers $m , n$ satisfying the inequality $a ^ { m } < a ^ { n } < b ^ { n } < b ^ { m }$, which of the following is correct? [3 points]
(1) $a < 1 < b , m > n$
(2) $a < 1 < b , m < n$
(3) $a < b < 1 , m < n$
(4) $1 < a < b , m > n$
(5) $1 < a < b , m < n$

For two exponential functions $f ( x ) = 4 ^ { x }$, $g ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$ with domain $\{ x \mid - 1 \leqq x \leqq 3 \}$, let $M$ be the maximum value of $f ( x )$ and $m$ be the minimum value of $g ( x )$. What is the value of $M m$? [3 points]
(1) 8
(2) 6
(3) 4
(4) 2
(5) 1

9. Let $f ( x ) = \ln x , 0 < a < b$. If $p = f ( \sqrt { a b } ) , q = f \left( \frac { a + b } { 2 } \right)$, $r = \frac { 1 } { 2 } ( f ( a ) + f ( b ) )$, then the correct relation is
A. $q = r < p$
B. $q = r > p$
C. $p = r < q$
D. $p = r > q$

7. Given the function $f ( x ) = 2 ^ { | x - m | } - 1$ defined on $\mathbb{R}$ (where $m$ is a real number), let $a = f \left( \log _ { 0.5 } 3 \right)$, $b = f \left( \log _ { 2 } 5 \right)$, $c = f ( 2m )$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $c < a < b$
(C) $a < c < b$
(D)

Given the function $\mathrm{f}(x) = 2^{|x-1|} - 1$ defined on $\mathbb{R}$ (where m is a real number) is an even function, let $\mathrm{a} = \mathrm{f}(\log_{0.5}3)$, $b = f(\log_2 5)$, $c = f(2m)$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $c < a < b$
(D) $c < b < a$

If $a > b > 1 , ~ 0 < c < 1$, then
(A) $a ^ { c } < b ^ { c }$
(B) $a b ^ { c } < b a ^ { c }$
(C) $a \log _ { b } c < b \log _ { a } c$
(D) $\log _ { a } c < \log _ { b } c$

Let $x, y, z$ be positive numbers such that $2 ^ { x } = 3 ^ { y } = 5 ^ { z }$, then
A. $2x < 3y < 5z$
B. $5z < 2x < 3y$
C. $3y < 5z < 2x$
D. $3y < 2x < 5z$

If $a > b$, then
A． $\ln ( a - b ) > 0$
B． $3 ^ { a } < 3 ^ { b }$
C．$a ^ { 3 } - b ^ { 3 } > 0$
D． $| a | > | b |$

Given $9 ^ { m } = 10 , a = 10 ^ { m } - 11 , b = 8 ^ { m } - 9$ , then
A. $a > 0 > b$
B. $a > b > 0$
C. $b > a > 0$
D. $b > 0 > a$

Below; the graphs of linear functions $f$, $g$ and $h$ are shown in Figure 1 on a rectangular coordinate plane divided into unit squares, and the derivatives of these functions are shown in Figure 2.
Accordingly; what is the correct ordering of $f ( 0 ) , g ( 0 )$ and $h ( 0 )$?
A) $\mathrm { f } ( 0 ) < \mathrm { h } ( 0 ) < \mathrm { g } ( 0 )$
B) $g ( 0 ) < f ( 0 ) < h ( 0 )$
C) $g ( 0 ) < h ( 0 ) < f ( 0 )$
D) $h ( 0 ) < f ( 0 ) < g ( 0 )$
E) $h ( 0 ) < g ( 0 ) < f ( 0 )$