119. The absolute minimum point of the function $f(x) = \dfrac{x^2 + 2x}{(x-1)^2}$ is at what distance from its vertical asymptote?
(1) $1$ (2) $\dfrac{4}{3}$ (3) $\dfrac{3}{2}$ (4) $2$

120. The mean value theorem applies to the function $y = \sqrt{21 - x^2 + 4x}$ on the interval $[6,\ 8]$. For the instantaneous rate of change to equal the average rate of change of this function, what value of $x$ is required?
(1) $4 + \sqrt{7}$ (2) $3 + 2\sqrt{7}$ (3) $2 + \dfrac{3}{2}\sqrt{7}$ (4) $2 + \dfrac{5}{2}\sqrt{7}$

122. For the function $f(x) = 2\sqrt{x} - \dfrac{3}{2\sqrt[3]{x^2}-1}$, which statement is correct?
(1) $f$ is increasing on $(1,\infty) \cup (0,1)$.
(2) $f$ is increasing on $(1,\infty)$ and $(0,1)$.
(3) $f$ is increasing on $(1,\infty)$ and decreasing on $(0,1)$.
(4) $f$ is decreasing on $(1,\infty)$ and increasing on $(0,1)$.

123. Consider the intervals on which $f(x) = \dfrac{x^4}{x^3 - 8}$ is strictly decreasing. What is the minimum total length of these intervals?
(1) $2$ (2) $\sqrt[4]{4}-1$ (3) $2\sqrt[4]{4}$ (4) $2(\sqrt[4]{4}-1)$

123-- Point $A(-1,1)$ is a relative extremum of the function $y=x^{2}|x|+3ax^{2}+b$. The value of $\dfrac{b}{a}$ is which of the following?
(1) $-3$ (2) $-\dfrac{1}{3}$ (3) $3$ (4) $\dfrac{1}{3}$

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

Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.

The minimum value of $x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}$ subject to $x_1 + x_2 + x_3 + x_4 = a$ and $x_1 - x_2 + x_3 - x_4 = b$ is
(a) $(a^{2} + b^{2})/4$
(b) $(a^{2} + b^{2})/2$
(c) $(a+b)^{2}/4$
(d) $(a+b)^{2}/2$

Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is
(a) $1 + 1 / \sqrt{2}$
(b) $2 - \sqrt{2}$
(c) $1 + \sqrt{2}$
(d) $( 1 + \sqrt{2} ) / 2$

Consider the function $f ( x ) = x ( x - 1 ) e ^ { 2 x }$ if $x \leq 0$ $f ( x ) = x ( 1 - x ) e ^ { - 2 x }$ if $x > 0$ Then $f ( x )$ attains its maximum value at
(a) $1 - 1 / \sqrt{2}$
(b) $1 + 1 / \sqrt{2}$
(c) $- 1 / \sqrt{2}$
(d) $1 / \sqrt{2}$

Let $f(x) = \dfrac{2x^2 + 3x + 1}{2x - 1}$. Find the maximum and minimum values of $f$ on $[2, 3]$.

If $xy = 1$, find the minimum value of $\dfrac{4}{4-x^2} + \dfrac{9}{9-y^2}$.

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e^{-x}$ is:
(A) $1/e$
(B) 1
(C) $1/2$
(D) $e$

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

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

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 $0 \leq x < \infty$.

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) $\alpha > 2$.