The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

Let $f(x) = ax^3 + bx^2 + cx + d$ be a strictly increasing function with $a > 0$. Define $g(x) = f'(x) - 6ax - 2b + 6a$. Then $g(x)$ is
(A) always negative (B) always positive (C) sometimes positive sometimes negative (D) always zero

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(A) $f ( x )$ is an increasing function on the real line
(B) $f ( x )$ is a decreasing function on the real line
(C) $f ( x )$ is increasing on $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$

Let $f ( x ) = \sin x + \alpha x , x \in \mathbb { R }$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha > 1$ or $\alpha < - 1$.
(B) $\alpha \geq 1$ or $\alpha \leq - 1$.
(C) $\alpha \geq 1$ or $\alpha < - 1$.
(D) $\alpha > 1$ or $\alpha \leq - 1$.

23. Let $f ( x ) = \int \operatorname { ex } ( x - 1 ) ( x - 2 ) d x$. Then $f$ decreases in the interval :
(A) $( \infty , - 2 )$
(B) $( - 2 , - 1 )$
(C) $( 1,2 )$
(D) $( 2 , + \infty )$

9. The length of a longest interval in which the function $3 \sin x - 4 \sin ^ { 3 } x$ is increasing, is
(A) $\pi / 3$
(B) $\pi / 2$
(C) $3 \sqcap / 3$
(D) $\Pi$

21. If $f ( x ) = \int _ { x } { } ^ { 2 ( x 2 + 1 ) } e ^ { - t 2 } d t$, then $f ( x )$ increases in:
(a) $\quad ( 2,2 )$
(b) no value of $x$
(c) $\quad ( 0 , ¥ )$
(d) $( - ¥ , 0 )$

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

Let the function $g : ( - \infty , \infty ) \rightarrow \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ be given by $g ( u ) = 2 \tan ^ { - 1 } \left( e ^ { u } \right) - \frac { \pi } { 2 }$. Then, $g$ is
(A) even and is strictly increasing in $(0 , \infty)$
(B) odd and is strictly decreasing in $( - \infty , \infty )$
(C) odd and is strictly increasing in $( - \infty , \infty )$
(D) neither even nor odd, but is strictly increasing in $( - \infty , \infty )$

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The function $f ^ { \prime } ( x )$ is
A) increasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and decreasing in $\left( - \frac { 1 } { 4 } , t \right)$
B) decreasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and increasing in $\left( - \frac { 1 } { 4 } , t \right)$
C) increasing in (-t, t)
D) decreasing in (-t, t)

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)
(A) $f ^ { \prime } ( 1 ) < 0$
(B) $f ( 2 ) < 0$
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

Let $f ( x ) = \lim _ { n \rightarrow \infty } \left( \frac { n ^ { n } ( x + n ) \left( x + \frac { n } { 2 } \right) \cdots \left( x + \frac { n } { n } \right) } { n ! \left( x ^ { 2 } + n ^ { 2 } \right) \left( x ^ { 2 } + \frac { n ^ { 2 } } { 4 } \right) \cdots \left( x ^ { 2 } + \frac { n ^ { 2 } } { n ^ { 2 } } \right) } \right) ^ { \frac { x } { n } }$, for all $x > 0$. Then
(A) $f \left( \frac { 1 } { 2 } \right) \geq f ( 1 )$
(B) $f \left( \frac { 1 } { 3 } \right) \leq f \left( \frac { 2 } { 3 } \right)$
(C) $f ^ { \prime } ( 2 ) \leq 0$
(D) $\frac { f ^ { \prime } ( 3 ) } { f ( 3 ) } \geq \frac { f ^ { \prime } ( 2 ) } { f ( 2 ) }$

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (i) (P)
[B] (II) (ii) (Q)
[C] (III) (iii) (R)
[D] (IV) (iv) (S)

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (ii) (R)
[B] (II) (iii) (S)
[C] (III) (iv) (P)
[D] (IV) (i) (S)

The function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ is an increasing function in
(1) $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
(3) $\left( 0 , \frac { \pi } { 2 } \right)$
(4) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

Let $f ( x ) = \sin ^ { 4 } x + \cos ^ { 4 } x$. Then, $f$ is an increasing function in the interval:
(1) $]\frac { 5 \pi } { 8 } , \frac { 3 \pi } { 4 } [$
(2) $]\frac { \pi } { 2 } , \frac { 5 \pi } { 8 } [$
(3) $]\frac { \pi } { 4 } , \frac { \pi } { 2 } [$
(4) $]0 , \frac { \pi } { 4 } [$