Consider the function $g$ defined on $[0; +\infty[$ by $g(t) = \frac{a}{b + \mathrm{e}^{-t}}$ where $a$ and $b$ are two real numbers. We know that $g(0) = 2$ and $\lim_{t \rightarrow +\infty} g(t) = 3$. The values of $a$ and $b$ are:
A. $a = 2$ and $b = 3$
B. $a = 4$ and $b = \frac{4}{3}$
C. $a = 4$ and $b = 1$
D. $a = 6$ and $b = 2$

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of $y = f ( x )$ and $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of the sum of the two constants $a + b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

When the graph of the exponential function $y = 5 ^ { x - 1 }$ passes through the two points $( a , 5 ) , ( 3 , b )$, find the value of $a + b$. [3 points]

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of the function $y = f ( x )$ and the function $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of $a + b$, the sum of the two constants $a , b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

14. If the function $f ( x ) = \left\{ \begin{array} { l } - x + 6 , x \leq 2 , \\ 3 + \log _ { a } x , x > 2 , \end{array} ( a > 0 \right.$ and $a \neq 1 )$ has range $[ 4 , + \infty )$, then the range of the real number $a$ is $\_\_\_\_$.

13. Given that the graph of function $f ( x ) = a x ^ { 3 } - 2 x$ passes through the point $( - 1,4 )$, then $a = $ $\_\_\_\_$ .

Let the function $f ( x ) = \mathrm { e } ^ { x } + a \mathrm { e } ^ { - x }$ ($a$ is a constant). If $f ( x )$ is an odd function, then $a =$ $\_\_\_\_$; if $f ( x )$ is an increasing function on $\mathbb { R }$, then the range of $a$ is $\_\_\_\_$.

1) Based on the information obtainable from the graph in Figure 2, show, with appropriate reasoning, that the function:
$$f ( x ) = \sqrt { 2 } - \frac { e ^ { x } + e ^ { - x } } { 2 } \quad x \in \mathbb { R }$$
adequately represents the profile of the platform for $x \in [ - a ; a ]$; also determine the value of the endpoints $a$ and $-a$ of the interval.

Ministry of Education, University and Research

To visualize the complete profile of the platform on which the bicycle will be able to move, several copies of the graph of the function $f(x)$ relating to the interval $[-a; a]$ are placed side by side, as shown in Figure 3.
[Figure]
Figure 3

33. If the line $x - 1 = 0$ is the directrix of the parabola $y 2 - k x + 8 = 0$ then one of the values of ... Powered By IITians k is :
(A) $1 / 8$
(B) 8
(C) 4
(D) $1 / 4$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \log _ { \sqrt { m } } \{ \sqrt { 2 } ( \sin x - \cos x ) + m - 2 \}$, for some $m$, such that the range of $f$ is $[ 0,2 ]$. Then the value of $m$ is $\_\_\_\_$.
(1) 5
(2) 3
(3) 2
(4) 4

Let $f: R \rightarrow R$ be defined $f(x) = ae^{2x} + be^x + cx$. If $f(0) = -1$, $f'(\log_e 2) = 21$ and $\int_0^{\log 4} (f(x) - cx)\, dx = \frac{39}{2}$, then the value of $|a + b + c|$ equals:
(1) 16
(2) 10
(3) 12
(4) 8

The difference between the maximum and minimum values of the function $f ( x ) = a ^ { \cos x }$, where $a > 0$ and $x$ is real, is 3 .
Find the sum of the possible values of $a$.