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 $\dfrac{4}{4-x^2} + \dfrac{9}{9-y^2}$.

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

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 $x = y ^ { 2 } + 1$.

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

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) $\alpha > 2$.

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$