Exercises

Subsection 3.5.4 Exercises

Subsubsection Exercises for § 3.5.1

Exercises — Stage 1

1

Identify every critical point and every singular point of $f(x)$ shown on the graph below. Which correspond to local extrema?

2

Identify every critical point and every singular point of $f(x)$ on the graph below. Which correspond to local extrema? Which correspond to global extrema over the interval shown?

3

Draw a graph $y=f(x)$ where $f(2)$ is a local maximum, but it is not a global maximum.

Exercises — Stage 2

4

Suppose $f(x)=\dfrac{x-1}{x^2+3}\text{.}$

Find all critical points.
Find all singular points.
What are the possible points where local extrema of $f(x)$ may exist?

Exercises — Stage 3

5

Below are a number of curves, all of which have a singular point at $x=2\text{.}$ For each, label whether $x=2$ is a local maximum, a local minimum, or neither.

6

Draw a graph $y=f(x)$ where $f(2)$ is a local maximum, but $x=2$ is not a critical point or an endpoint.

7

\begin{equation*} f(x)=\sqrt{\left|(x-5)(x+7)\right|} \end{equation*}

Find all critical points and all singular points of $f(x)\text{.}$ You do not have to specify whether a point is critical or singular.

8

Suppose $f(x)$ is the constant function $f(x)=4\text{.}$ What are the critical points and singular points of $f(x)\text{?}$ What are its local and global maxima and minima?

Subsubsection Exercises for § 3.5.2

Exercises — Stage 1

1

Sketch a function $f(x)$ such that:

$f(x)$ is defined over all real numbers
$f(x)$ has a global max but no global min.

2

Sketch a function $f(x)$ such that:

$f(x)$ is defined over all real numbers
$f(x)$ is always positive
$f(x)$ has no global max and no global min.

3

Sketch a function $f(x)$ such that:

$f(x)$ is defined over all real numbers
$f(x)$ has a global minimum at $x=5$
$f(x)$ has a global minimum at $x=-5\text{,}$ too.

Exercises — Stage 2

4

$f(x)=x^2+6x-10\text{.}$ Find all global extrema on the interval $[-5,5]$

5

$f(x)=\dfrac{2}{3}x^3-2x^2-30x+7\text{.}$ Find all global extrema on the interval $[-4,0]\text{.}$

Subsubsection Exercises for § 3.5.3

Exercises — Stage 1

For Questions 3.5.4.1 through 3.5.4.3, the quantity to optimize is already given to you as a function of a single variable.

For Questions 3.5.4.4 and 3.5.4.5, you can decide whether a critical point is a local extremum by considering the derivative of the function.

For Questions 3.5.4.6 through 3.5.4.13, you will have to find an expression for the quantity you want to optimize as a function of a single variable.

1 (✳)

Find the global maximum and the global minimum for $f(x)=x^5 - 5x + 2$ on the interval $[-2,0]\text{.}$

2 (✳)

Find the global maximum and the global minimum for $f(x)=x^5 - 5x - 10$ on the interval $[0,2]\text{.}$

3 (✳)

Find the global maximum and the global minimum for $f(x)=2x^3 - 6x^2 - 2$ on the interval $[1,4]\text{.}$

4 (✳)

Consider the function $h(x)=x^3-12x+4\text{.}$ What are the coordinates of the local maximum of $h(x)\text{?}$ What are the coordinates of the local minimum of $h(x)\text{?}$

5 (✳)

Consider the function $h(x)=2x^3-24x+1\text{.}$ What are the coordinates of the local maximum of $h(x)\text{?}$ What are the coordinates of the local minimum of $h(x)\text{?}$

6 (✳)

You are in a dune buggy at a point $P$ in the desert, 12 km due south of the nearest point $A$ on a straight east-west road. You want to get to a town $B$ on the road $18$ km east of $A\text{.}$ If your dune buggy can travel at an average speed of 15 km/hr through the desert and 30 km/hr along the road, towards what point $Q$ on the road should you head to minimize your travel time from $P$ to $B\text{?}$

7 (✳)

A closed three dimensional box is to be constructed in such a way that its volume is 4500 cm${}^3\text{.}$ It is also specified that the length of the base is 3 times the width of the base. Find the dimensions of the box that satisfies these conditions and has the minimum possible surface area. Justify your answer.

8 (✳)

A closed rectangular container with a square base is to be made from two different materials. The material for the base costs $5 per square metre, while the material for the other five sides costs $1 per square metre. Find the dimensions of the container which has the largest possible volume if the total cost of materials is $72.

9 (✳)

Find a point $X$ on the positive $x$--axis and a point $Y$ on the positive $y$--axis such that (taking $O=(0,0)$)

The triangle $XOY$ contains the first quadrant portion of the unit circle $x^2+y^2=1$ and
the area of the triangle $XOY$ is as small as possible.

A complete and careful mathematical justification of property 3.5.4.9.i is required.

10 (✳)

A rectangle is inscribed in a semicircle of radius $R$ so that one side of the rectangle lies along a diameter of the semicircle. Find the largest possible perimeter of such a rectangle, if it exists, or explain why it does not. Do the same for the smallest possible perimeter.

11 (✳)

Find the maximal possible volume of a cylinder with surface area $A\text{.}$ ¹²Food is often packaged in cylinders, and companies wouldn't want to waste the metal they are made out of. So, you might expect the dimensions you find in this problem to describe a tin of, say, cat food. Read here about why this isn't the case.

12 (✳)

What is the largest possible area of a window, with perimeter $P\text{,}$ in the shape of a rectangle with a semicircle on top (so the diameter of the semicircle equals the width of the rectangle)?

13 (✳)

Consider an open-top rectangular baking pan with base dimensions $x$ centimetres by $y$ centimetres and height $z$ centimetres that is made from $A$ square centimetres of tin plate. Suppose $y = px$ for some fixed constant $p\text{.}$

Find the dimensions of the baking pan with the maximum capacity (i.e., maximum volume). Prove that your answer yields the baking pan with maximum capacity. Your answer will depend on the value of $p\text{.}$
Find the value of the constant $p$ that yields the baking pan with maximum capacity and give the dimensions of the resulting baking pan. Prove that your answer yields the baking pan with maximum capacity.

Exercises — Stage 3

14 (✳)

Let $f(x)=x^x$ for $x \gt 0\text{.}$

Find $f'(x)\text{.}$
At what value of $x$ does the curve $y=f(x)$ have a horizontal tangent line?
Does the function $f$ have a local maximum, a local minimum, or neither of these at the point $x$ found in part 3.5.4.14.b?

15 (✳)

A length of wire is cut into two pieces, one of which is bent to form a circle, the other to form a square. How should the wire be cut if the area enclosed by the two curves is maximized? How should the wire be cut if the area enclosed by the two curves is minimized? Justify your answers.