- If f′ = 0
, is it necessary that x is either a local minimum or local
maximum? Hint: Consider f =
- A continuous function f defined on is to be maximized. It was shown
above in Theorem
5.7.2 that if the maximum value of f occurs at x ∈
and if f is differentiable there, then f′ = 0
. However, this theorem does
not say anything about the case where the maximum of f occurs at either a
or b. Describe how to find the point of where
f achieves its maximum.
Does f have a maximum? Explain.
- Show that if the maximum value of a function f differentiable on occurs
at the right endpoint, then for all
h > 0,f′
h ≥ 0. This is an example of a
variational inequality. Describe what happens if the maximum occurs at the
left end point and give a similar variational inequality. What is the situation
- Let y = xx for x ∈ (0,∞). Find y′
- Find the maximum and minimum values and the values of x where these are
achieved for the function f =
x + .
- A piece of wire of length L is to be cut in two pieces. One piece is bent into the
shape of an equilateral triangle and the other piece is bent to form a square.
How should the wire be cut to maximize the sum of the areas of the two
shapes? How should the wire be bent to minimize the sum of the areas of the
two shapes? Hint: Be sure to consider the case where all the wire is devoted
to one of the shapes separately. This is a possible solution even though the
derivative is not zero there.
- Lets find the point on the graph of y = which is closest to
way to do it is to observe that a typical point on the graph is of the form
and then to minimize the function
the derivative of f yields x +
x3 and setting this equal to 0 leads to the
solution, x = 0. Therefore, the point closest to is
. Now lets do it
another way. Lets use y = to write
x2 = 4y. Now for on the graph, it
follows it is of the form
. Therefore, minimize f = 4
Take the derivative to obtain 2 + 2y which requires y = −1. However, on this
graph, y is never negative. What on earth is the problem?
- Find the dimensions of the largest rectangle that can be inscribed in the
- A function f, is said to be odd if f =
−f and a function is said to be
. Show that if f is even, then f′ is odd and if f is odd,
then f′ is even. Sketch the graph of a typical odd function and a typical even
- Find the point on the curve, y = which is closest to
- A street is 200 feet long and there are two lights located at the ends of
the street. One of the lights is times as bright as the other. Assuming
the brightness of light from one of these street lights is proportional to the
brightness of the light and the reciprocal of the square of the distance from
the light, locate the darkest point on the street.
- Find the area of the smallest rectangle which can be circumscribed about a
circle of radius 4 inches.
- Find the maximum and minimum values for the following functions defined
on the given intervals.
(a) x3 − 3x2 + x − 7,
x2 − x
+ 4x −
- A cylindrical can is to be constructed to hold 30 cubic inches. The top and bottom
of the can are constructed of a material costing one cent per square inch and the
sides are constructed of a material costing 2 cents per square inch. Find the
minimum cost for such a can.
- Two positive numbers sum to 8. Find the numbers if their product is to be as large
- The ordered pair is on the ellipse
x2 + 4y2 = 4. Form the rectangle which has
as one end of a diagonal and
at the other end. Find the rectangle of
this sort which has the largest possible area.
- A rectangle is inscribed in a circle of radius r. Find the formula for the rectangle of
this sort which has the largest possible area.
- A point is picked on the ellipse x2 + 4y2 = 4 which is in the first quadrant.
Then a line tangent to this point is drawn which intersects the x axis
at a point x1 and the y axis at the point y1. The area of the triangle
formed by the y axis, the x axis, and the line just drawn is thus . Out
of all possible triangles formed in this way, find the one with smallest
- Find maximum and minimum values if they exist for the function f =
x > 0.
- Describe how you would find the maximum value of the function f =
x ∈ if it exists.
Hint: You might want to use a calculator to graph this and
get an idea what is going on.
- A rectangular beam of height h and width w is to be sawed from a circular log of
radius 1 foot. Find the dimensions of the strongest such beam assuming the
strength is of the form kh2w. Here k is some constant which depends on the type of
- A farmer has 600 feet of fence with which to enclose a rectangular piece of land
that borders a river. If he can use the river as one side, what is the largest area that
he can enclose.
- An open box is to be made by cutting out little squares at the corners of a
rectangular piece of cardboard which is 20 inches wide and 40 inches long and then
folding up the rectangular tabs which result. What is the largest possible volume
which can be obtained?
- A feeding trough is to be made from a rectangular piece of metal which is 3 feet
wide and 12 feet long by folding up two rectangular pieces of dimension one foot by
12 feet. What is the best angle for this fold?
- Find the dimensions of the right circular cone which has the smallest area given the
volume is 30π cubic inches. The volume of the right circular cone is
the area of the cone is πr.
- A wire of length 10 inches is cut into two pieces, one of length x and the
other of length 10 − x. One piece is bent into the shape of a square and
the other piece is bent into the shape of a circle. Find the two lengths
such that the sum of the areas of the circle and the square is as large as
possible. What are the lengths if the sum of the two areas is to be as small as
- A hiker begins to walk to a cabin in a dense forest. He is walking on a road which
runs from East to West and the cabin is located exactly one mile north of a point
two miles down the road. He walks 5 miles per hour on the road but only 3 miles
per hour in the woods. Find the path which will minimize the time it takes for him
to get to the cabin.
- A park ranger needs to get to a fire observation tower which is one mile
from a long straight road in a dense forest. The point on the road closest
to the observation tower is 10 miles down the road on which the park
ranger is standing. Knowing that he can walk at 4 miles per hour on the
road but only one mile per hour in the forest, how far down the road
should he walk before entering the forest, in order to minimize the travel
- A refinery is on a straight shore line. Oil needs to flow from a mooring place for oil
tankers to this refinery. Suppose the mooring place is two miles off shore from a
point on the shore 8 miles away from the refinery which is also on the shore and
that it costs five times as much to lay pipe under water than above the ground.
Describe the most economical route for a pipeline from the mooring place to the
- Two hallways, one 5 feet wide and the other 6 feet wide meet. It is desired to carry
a ladder horizontally around the corner. What is the longest ladder which can be
carried in this way? Hint: Consider a line through the inside corner which extends
to the opposite walls. The shortest such line will be the length of the longest
- A window is to be constructed for the wall of a church which is to consist of a
rectangle of height b surmounted by a half circle of radius a. Suppose the total
perimeter of the window is to be no more than 4π + 8 feet. Find the dimensions of
the window which will admit the most light.
- You know limx→∞lnx = ∞. Show that if α > 0, then limx→∞ = 0
- ∗ A parabola opens down. The vertex is at the point and the parabola
x axis at the points and
. A tangent line to the
parabola is drawn in the first quadrant which has the property that the triangle
formed by this tangent line and the x and y axes has smallest possible area. Find a
relationship between a and b such that the normal line to the point of
tangency passes through
. Also determine what kind of triangle this
- Show that for r a rational number and y = xr, it must be the case that if this
function is differentiable, then y′ = rxr−1. This was shown in more generality, but
use the chain rule to verify this directly.
Now let g =
. Find where g is continuous and differentiable if
- Verify the last two lines of the table for derivatives given above, those involving the
hyperbolic functions cosh