Let $f$ be the function with derivative defined by $f ^ { \prime } ( x ) = 2 + ( 2 x - 8 ) \sin ( x + 3 )$.
How many points of inflection does the graph of $f$ have on the interval $0 < x < 9$?
(A) One
(B) Two
(C) Three
(D) Four

19. If $f ^ { \prime \prime } ( x ) = x ( x + 1 ) ( x - 2 ) ^ { 2 }$, then the graph of $f$ has inflection points when $x =$
(A) - 1 only
(B) 2 only
(C) - 1 and 0 only
(D) -1 and 2 only
(E) $- 1,0$, and 2 only

2. The function $g$ is defined for $x > 0$ with $g ( 1 ) = 2 , g ^ { \prime } ( x ) = \sin \left( x + \frac { 1 } { x } \right)$, and $g ^ { \prime \prime } ( x ) = \left( 1 - \frac { 1 } { x ^ { 2 } } \right) \cos \left( x + \frac { 1 } { x } \right)$.
(a) Find all values of $x$ in the interval $0.12 \leq x \leq 1$ at which the graph of $g$ has a horizontal tangent line.
(b) On what subintervals of $( 0.12,1 )$, if any, is the graph of $g$ concave down? Justify your answer.
(c) Write an equation for the line tangent to the graph of $g$ at $x = 0.3$.
(d) Does the line tangent to the graph of $g$ at $x = 0.3$ lie above or below the graph of $g$ for $0.3 < x < 1$ ? Why?

$t$	0	2	4	6	8	10	12
$P ( t )$	0	46	53	57	60	62	63

[Figure]

Let $g$ be a function with first derivative given by $g ^ { \prime } ( x ) = \int _ { 0 } ^ { x } e ^ { - t ^ { 2 } } d t$. Which of the following must be true on the interval $0 < x < 2$ ?
(A) $g$ is increasing, and the graph of $g$ is concave up.
(B) $g$ is increasing, and the graph of $g$ is concave down.
(C) $g$ is decreasing, and the graph of $g$ is concave up.
(D) $g$ is decreasing, and the graph of $g$ is concave down.
(E) $g$ is decreasing, and the graph of $g$ has a point of inflection on $0 < x < 2$.

For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) $(-0.418, 0.418)$ only
(B) $( - 1,1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409,1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0,1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409,0 )$, and $( 1.354,1.5 )$

The graph of $f ^ { \prime \prime }$, the second derivative of $f$, is shown above for $- 2 \leq x \leq 4$. What are all intervals on which the graph of the function $f$ is concave down?
(A) $-1 < x < 1$
(B) $0 < x < 2$
(C) $1 < x < 3$ only
(D) $-2 < x < -1$ only
(E) $-2 < x < -1$ and $1 < x < 3$

Let $f$ be a function that is twice differentiable on $- 2 < x < 2$ and satisfies the conditions in the table above. If $f ( x ) = f ( - x )$, what are the $x$-coordinates of the points of inflection of the graph of $f$ on $- 2 < x < 2$ ?

\cline{2-3} \multicolumn{1}{c|}{}	$0 < x < 1$	$1 < x < 2$
$f ( x )$	Positive	Negative
$f ^ { \prime } ( x )$	Negative	Negative
$f ^ { \prime \prime } ( x )$	Negative	Positive

(A) $x = 0$ only
(B) $x = 1$ only
(C) $x = 0$ and $x = 1$
(D) $x = - 1$ and $x = 1$
(E) There are no points of inflection on $- 2 < x < 2$.

For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) (-0.418, 0.418) only
(B) $( - 1, 1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409, 1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0, 1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409, 0 )$, and $( 1.354, 1.5 )$

The derivative of a function $f$ is increasing for $x < 0$ and decreasing for $x > 0$. Which of the following could be the graph of $f$ ?
(A), (B), (C), (D), (E) [graphs shown in figures above]

Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$.
(a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False?
(b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False?
(c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.

Let $a$ be a point in the domain of a continuous real valued function $f$. One says that $f$ has a flex point at $a$ if we can find a small interval $(a - \epsilon , a + \epsilon)$ in the domain of $f$ such that the following happens: (i) for all $x$ in the open interval $(a - \epsilon , a)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and, (ii) for all $x$ in the open interval $(a , a + \epsilon)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and opposite to the sign of $f ^ { \prime \prime } ( x )$ in $(a - \epsilon , a)$.
Statements
(29) If $f$ is a cubic polynomial with a local maximum at $x = p$ and a local minimum at $x = q$, then $f$ has a unique flex point at $x = \frac { p + q } { 2 }$. (30) If $f ^ { \prime \prime } ( a ) = 0$ then $f$ must have a flex point at $a$. (31) Let $f ( x ) = x ^ { 2 }$ for $x \geq 0$ and $f ( x ) = - x ^ { 2 }$ for $x < 0$. Then $f$ has no flex points. (32) If $f ^ { \prime }$ is monotonic on an open interval $I$, then $f$ cannot have a flex point in $I$.

How many integers $a$ are there such that the curve $y = a x ^ { 2 } - 2 \sin 2 x$ has an inflection point? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

120- For which set of values of $a$, the curve $y = x^4 + ax^3 + \dfrac{3}{2}x^2$ is always concave up?
(1) $-1 < a < 1$ (2) $-1 < a < 2$ (3) $-2 < a < 1$ (4) $-2 < a < 2$

118. From the relation $x^2y - y^2 - 2\sqrt{x} + 4 = 0$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1, 2)$ is which of the following?
(1) $\dfrac{7}{6}$ (2) $\dfrac{8}{6}$ [6pt] (3) $\dfrac{11}{6}$ (4) $\dfrac{13}{6}$

120- What is the length of the inflection point of the graph of $y = (5-x)\sqrt[3]{x^2}$?
(1) $-1$ (2) zero (3) $1$ (4) $2$

122. For the graph of $f(x) = \cos^2 x - 2\cos x$; $x \in [0, 2\pi]$, at which base point is the inflection point and local minimum?
(1) $\left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ (2) $\left(\pi, \dfrac{4\pi}{3}\right)$ (3) $\left(\dfrac{2\pi}{3}, \pi\right)$ (4) $\left(\dfrac{4\pi}{3}, \dfrac{3\pi}{2}\right)$

19 -- For how many positive and negative integer values of $k$, does the inflection point of $y = kx^3 + (k+1)x^2$ lie in the second quadrant?
(1) $1$ (2) $2$ (3) more than $2$ (4) zero

If $f : \mathbb { R } \rightarrow \mathbb { R }$ is a twice differentiable function such that $f ^ { \prime \prime } ( x ) > 0$ for all $x \in \mathbb { R }$, and $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } , f ( 1 ) = 1$, then
[A] $f ^ { \prime } ( 1 ) \leq 0$
[B] $0 < f ^ { \prime } ( 1 ) \leq \frac { 1 } { 2 }$
[C] $\frac { 1 } { 2 } < f ^ { \prime } ( 1 ) \leq 1$
[D] $f ^ { \prime } ( 1 ) > 1$

Let $f ( x ) = 3 ^ { \left( x ^ { 2 } - 2 \right) ^ { 3 } + 4 } , \mathrm { x } \in R$. Then which of the following statements are true? $P : x = 0$ is a point of local minima of $f$ $Q : x = \sqrt { 2 }$ is a point of inflection of $f$ $R : f ^ { \prime }$ is increasing for $x > \sqrt { 2 }$
(1) Only $P$ and $Q$
(2) Only $P$ and $R$
(3) Only $Q$ and $R$
(4) All $P , Q$ and $R$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$