The function $x ^ { 2 } \log _ { e } x$ in the interval $( 0,2 )$ has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $f ( x )$ is decreasing on $( - 1,1 )$ and has a local minimum at $x = 1$
(B) $f ( x )$ is increasing on $( - 1,1 )$ and has a local maximum at $x = 1$
(C) $f ( x )$ is increasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$
(D) $f ( x )$ is decreasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$

Let f be a function defined on $\mathbf { R }$ (the set of all real numbers) such that $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2010 ( \mathrm { x } - 2009 ) ( \mathrm { x } - 2010 ) ^ { 2 } ( \mathrm { x } - 2011 ) ^ { 3 } ( \mathrm { x } - 2012 ) ^ { 4 }$, for all $\mathrm { x } \in \mathbf { R }$.
If $g$ is a function defined on $\mathbf { R }$ with values in the interval $( 0 , \infty )$ such that
$$\mathrm { f } ( \mathrm { x } ) = \ell n ( \mathrm {~g} ( \mathrm { x } ) ) \text {, for all } \mathrm { x } \in \mathbf { R } \text {, }$$
then the number of points in $\mathbf { R }$ at which $g$ has a local maximum is

Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then
(A) $f$ has a local minimum at $x = 2$
(B) $f$ has a local maximum at $x = 2$
(C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$
(D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$

Let $$f(x) = \frac{\sin\pi x}{x^2}, \quad x > 0$$
Let $x_1 < x_2 < x_3 < \cdots < x_n < \cdots$ be all the points of local maximum of $f$ and $y_1 < y_2 < y_3 < \cdots < y_n < \cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct?
(A) $x_1 < y_1$
(B) $x_{n+1} - x_n > 2$ for every $n$
(C) $x_n \in \left(2n, 2n + \frac{1}{2}\right)$ for every $n$
(D) $|x_n - y_n| > 1$ for every $n$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define $$F(x) = \int_0^x f(t)\,dt, \quad x > 0$$
Then which of the following options is/are correct?
(A) $F$ has a local minimum at $x = 1$
(B) $F$ has a local maximum at $x = 2$
(C) $F$ has two local maxima and one local minimum in $(0, \infty)$
(D) $F(x) \neq 0$ for all $x \in (0, 5)$

Let the function $f: (0, \pi) \rightarrow \mathbb{R}$ be defined by $$f(\theta) = (\sin\theta + \cos\theta)^{2} + (\sin\theta - \cos\theta)^{4}$$ Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_{1}\pi, \ldots, \lambda_{r}\pi\}$, where $0 < \lambda_{1} < \cdots < \lambda_{r} < 1$. Then the value of $\lambda_{1} + \cdots + \lambda_{r}$ is $\_\_\_\_$

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $2m_1 + 3n_1 + m_1 n_1$ is ____.

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $6m_2 + 4n_2 + 8m_2 n_2$ is ____.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $2 m _ { 1 } + 3 n _ { 1 } + m _ { 1 } n _ { 1 }$ is $\_\_\_\_$.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $6 m _ { 2 } + 4 n _ { 2 } + 8 m _ { 2 } n _ { 2 }$ is $\_\_\_\_$.

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \begin{cases} \frac { 6 x + \sin x } { 2 x + \sin x } & \text { if } x \neq 0 \\ \frac { 7 } { 3 } & \text { if } x = 0 \end{cases}$$
Then which of the following statements is (are) TRUE?

(A)	The point $x = 0$ is a point of local maxima of $f$
(B)	The point $x = 0$ is a point of local minima of $f$
(C)	Number of points of local maxima of $f$ in the interval $[ \pi , 6 \pi ]$ is 3
(D)	Number of points of local minima of $f$ in the interval $[ 2 \pi , 4 \pi ]$ is 1

Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is:
(1) $2 \sqrt { 2 }$
(2) 3
(3) - 3
(4) $- 2 \sqrt { 2 }$

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in R$, then
(1) $S_1 = \{-2\}$; $S_2 = \{0, 1\}$
(2) $S_1 = \{-1\}$; $S_2 = \{0, 2\}$
(3) $S_1 = \{-2, 0\}$; $S_2 = \{1\}$
(4) $S_1 = \{-2, 1\}$; $S_2 = \{0\}$

If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly
(1) Two irrational and two rational numbers
(2) Four rational numbers
(3) Two irrational and one rational number
(4) Four irrational numbers

Let $f ( x )$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim _ { x \rightarrow 0 } \left( 2 + \frac { f ( x ) } { x ^ { 3 } } \right) = 4$, then which one of the following is not true?
(1) $f$ is an odd function
(2) $f ( 1 ) - 4 f ( - 1 ) = 4$
(3) $x = 1$ is a point of local minimum and $x = - 1$ is a point of local maximum
(4) $x = 1$ is a point of local maxima of $f$

Let a function $f : [ 0,5 ] \rightarrow R$ be continuous, $f ( 1 ) = 3$ and $F$ be defined as: $F ( x ) = \int _ { 1 } ^ { x } t ^ { 2 } g ( t ) d t$, where $g ( t ) = \int _ { 1 } ^ { t } f ( u ) d u$. Then for the function $F ( x )$, the point $x = 1$ is:
(1) a point of local minima
(2) not a critical point
(3) a point of local maxima
(4) a point of inflection

If $x = 1$ is a critical point of the function $f(x) = (3x^2 + ax - 2 - a)e^x$, then
(1) $x = 1$ and $x = -\frac{2}{3}$ are local minima of $f$
(2) $x = 1$ and $x = -\frac{2}{3}$ is a local maxima of $f$
(3) $x = 1$ is a local maxima and $x = -\frac{2}{3}$ is a local minima of $f$
(4) $x = 1$ is a local minima and $x = -\frac{2}{3}$ are local maxima of $f$

Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a
(1) local maximum at $x = - \frac { 3 } { 4 }$
(2) local minimum at $x = - \frac { 3 } { 4 }$
(3) local maximum at $x = \frac { 3 } { 4 }$
(4) local minimum at $x = \frac { 3 } { 4 }$

The sum of all the local minimum values of the twice differentiable function $f : R \rightarrow R$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - \frac { 3 f ^ { \prime \prime } ( 2 ) } { 2 } x + f ^ { \prime \prime } ( 1 )$ is:
(1) - 22
(2) 5
(3) - 27
(4) 0

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { t ^ { 2 } - 5 t + 4 } { 2 + e ^ { t } } d t$, then the ordered pair $( m , n )$ is equal to
(1) $( 2,3 )$
(2) $( 3,2 )$
(3) $( 2,2 )$
(4) $( 3,4 )$