Deck 15: Optimization- Local and Global Extrema

Question 1

Suppose that $f ( x , y ) = x ^ { 2 } + 3 x y + y ^ { 2 }$ Find an equation of the tangent plane to the graph of f at the point (2, 2).

A) $z = 20 + 10 ( x - 2 ) + 10 ( y - 2 )$
B) $z = 20 - 10 ( x - 2 ) - 10 ( y - 2 )$
C) $z = 20 - 10 ( x - 2 ) + 10 ( y - 2 )$
D) $z = 20 + 10 x + 10 y$

Answer 1

$z = 20 + 10 ( x - 2 ) + 10 ( y - 2 )$

Question 2

Let Determine all local maxima, minima, and saddle points.Are the local extrema also global extrema?

Answer 2

(0, 0)is a saddle point; (-3, -3)is a local maximum.There is no global extremum.

Question 3

Find the critical points of and classify each as maximum, minimum or saddle.

Answer 3

The point (0, 0)is a saddle point.
The point is a local minimum.

Question 4

Level curves of f(x, y)are shown in the figure below.(Darker shades indicate regions with lower levels.)
Determine if is positive, negative or zero.

Question 5

The function $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 6 y$ has a saddle point at (-1, 12).Which of the following is a sketch of the level curves of f near this point?

A)
B)

Question 6

Level curves of f(x, y)are shown in the figure below.(Darker shades indicate regions with lower levels.) Is the point (-1, 2)a local maximum, a local minimum, or a saddle point of f, or is it none of these?

A)Saddle point.
B)Local minimum.
C)Local maximum.
D)None of these.

Question 7

Consider the function Check that (0,0)is a critical point of f and classify it as a local minimum, local maximum or saddle point.

Question 8

Let $h ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 x y + 5$ Which figure best represents the level curves of this function?

A)
B)
C)
D)

Question 9

The function f(x, y)has a local maximum at (-1, 1). What can you say (if anything)about the values of $f _ { x y } ( - 1,1 )$ ?

A)Nothing.
B)It is undefined or equal to zero.
C)It cannot be greater than zero.
D)It is equal to zero.

Question 10

The contour diagram of f is shown below. Find and classify the critical points.
Describe possible gradient vectors of f at points C, D and E.

Question 11

Suppose that Find and classify the critical point(s)as local maxima, local minima, or saddle points.

Question 12

Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 2 y ^ { 2 } - b x$ , where a, b are any numbers with a $\neq$ 2.
Find the critical point of f.(Express your answer in terms of the constants a and b.)

Question 13

Is (0, 0)a critical point of the following function? $f ( x , y ) = \cos x \cos y$

A)No.
B)Yes: (global)maximum.
C)Yes: local minimum.
D)Yes: (global)minimum.

Question 14

Find all the critical points of $f ( x , y ) = x ^ { 3 } - 3 x + y ^ { 2 } - 4 y$ and classify each as maximum, minimum, or saddle point. Select all possible choices.

A)The point (1, 2)is a relative minimum.
B)The point (-1, -2)is a saddle point.
C)The point (-1, 2)is a saddle point.
D)The point (1, 2)is a relative maximum.
E)The point (1, -2)is a relative maximum.

Question 15

Suppose that $f ( x , y ) = x ^ { 2 } + 3 x y + y ^ { 2 }$ Find a normal vector to the tangent plane of f at the point (1, 1).Select all that apply.

A) $5 \vec { i } + 5 \vec { j } - \vec { k }$
B) $5 \vec { i } + 5 \vec { j } + \vec { k }$
C) $- 5 \vec { i } - 5 \vec { j } + \vec { k }$
D) $- 5 \vec { i } + 5 \vec { j } + \vec { k }$

Question 16

Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 4 y ^ { 2 } - b x$ , where a, b are any positive numbers with a $\gt$ 4.
Find the minimum value of a such that the critical point will be a saddle point.

Question 17

The contour diagram of f is shown below.Which of the points A, B, C, D, and E appear to be critical points? Select all that apply.

A)A
B)B
C)C
D)D
E)E

Question 18

Suppose that (1, -4)is a critical point of a smooth function f(x, y)with Find the quadratic approximation of f at (1, -4).

Question 19

Suppose that (0, -2)is a critical point of a smooth function f(x, y)with What can you conclude about the behavior of the function f near (0, -2)?

Question 20

Find all the critical points of the function Classify these critical points as local maxima, local minima, or saddle points.

Question 21

Determine the nature of the critical points of the function $f ( x , y ) = 3 x ^ { 5 } y + x y ^ { 5 } + x y$

A)f has a saddle point at (1, 0).
B)f has a local maximum at (0, 0).
C)f has a saddle point at (0, 1).
D)f has a saddle point at (0, 0).
E)f has a local minimum at (0, 0).

Question 22

Consider the four points A = (1, 0), B = (2, 3), C = (3, 5)and D = (4, 3)in the xy-plane.
Find the values of a, b and c to determine the parabola of best fit, for these points.
(The parabola of best fit minimizes the sum of the squares of the vertical distances from each point to the parabola.)

Question 23

Consider the four points A = (1, 0), B = (2, 2), C = (3, 5)and D = (4, 3)in the xy-plane.
Find a and b in the line of best fit y = ax + b for these points.
(The line of best fit minimizes the sum of the squares of the vertical distances from each point to the line.)

Question 24

Consider the diagram shown below, which shows gradient vectors of a function f(x, y). Which is less: f(A)or f(C)?

Question 25

The point (-2, 1)is a critical point of Classify it either as a local minimum, local maximum, or saddle point.

Question 26

Suppose that Find and classify (as local maxima, minima, or saddle points)all critical points of f.

Question 27

Without calculating the discriminant, explain using the contour diagram for why f has a saddle point at (0, 0).

Question 28

Find the critical points of Do this by setting and optimizing subject to the constraint

Question 29

What do the second derivatives tell you about the graph of $f ( x , y ) = 7 x ^ { 2 } + 1000 x y + 7 y ^ { 2 } - 3 x - 12 y + 18 ?$ Select all that apply.

A)Any critical point must be a saddle point.
B)There is a local minimum.
C)There are no local minima.
D)The function has a local maximum.
E)There are at least two local minima.

Question 30

Find a and b so that has a critical point at (1, 6).

Question 31

The function where a and b are constants is sometimes referred to as a "bump function" and is used to construct functions which take on maximum values at certain points.Show that f(x, y)has a maximum at (a, b).

Question 32

A zoo is designing a giant bird cage consisting of a cylinder of radius r feet and height h feet with a hemisphere on top (no bottom).The material for the hemisphere costs $20 per square foot and the material for the cylindrical sides costs $10 per square foot; the zoo has a budget of $5120.Find the values of r and h giving the birds the greatest space inside assuming the zoo stays within its budget.

Question 33

Consider the function Determine all the local maximum, minimum and saddle points in the region

Question 34

Find the critical point of Do this by setting and optimizing subject to the constraint What are the global maximum and minimum values of f? Give your answer to 4 decimal places.

Question 35

The Perfect House company produces two types of bathtub, the Hydro Deluxe model and the Singing Bird model.The company noticed that demand and prices are related.In particular,
for Hydro Deluxe: demand = 1900 - price of Hydro Deluxe + price of Singing Bird
for Singing Bird: demand = 1450 + price of Hydro Deluxe -2(price of Singing Bird).
The costs of manufacturing the Hydro Deluxe and Singing Bird are $500 and $300 per unit respectively.Determine the price of each model that gives the maximum profit.

Question 36

Find a point on the surface x-yz = 14 that is closest to the origin.

Question 37

If C is a circle in the plane, and if f(x, y)is differentiable and is not constant when constrained to C, then there must be at least one point on C where $\operatorname { grad } f$ is perpendicular to C.

Question 38

Describe the shape of the graph of $f ( x , y ) = 3 x ^ { 2 } + 3 y ^ { 2 } - 3 x - 12 y + 18$

A)Paraboloid
B)Elliptic paraboloid
C)Hyperbolic paraboloid
D)None of the above

Question 39

A company has $250,000 to spend on labor and raw materials.Let L be the quantity of labor and R be the quantity of raw materials.The production output P of the company is cRL (here c is a positive constant).Suppose that each unit of labor costs $6000 and the unit price of raw materials is $2000.
Find the ratio of R to L that maximizes P.

Question 40

A company has two manufacturing plants which manufacture the same item.Suppose the cost function is given by where q₁ and q₂ are the quantities (measured in thousands)produced in each plant.The total demand q₁ + q₂ is related to the price, p, by How much should each plant produce in order to maximize the company's profit?

Question 41

(a)Let $f ( x , y ) = x ^ { 2 } - 4 x y + 2 y ^ { 2 }$ .Find the maximum and minimum values of f on the curve $x ^ { 2 } + 2 y ^ { 2 } = 1$
(b)Use the results of part (a)to find the maximum and minimum values of $h ( x , y ) = e ^ { x ^ { 2 } - 4 x y + 2 y ^ { 2 } }$ on the curve $x ^ { 2 } + 2 y ^ { 2 } = 1$ Explain your work.

Question 42

The owner of a jewelry store has to decide how to allocate a budget of $540,000.He notices that the earnings of the company depend on investment in inventory x₁ (in thousands of dollars)and expenditure x₂ on advertising (in thousands of dollars)according to the function How should the owner allocate the $540,000 between inventory and advertising to maximize his profit?

Question 43

Find the saddle point of .

Question 44

Suppose that you want to find the maximum and minimum values of subject to the constraint x + 4y = 3.
Use the method of Lagrange multipliers to find the exact location(s)of any extrema.

Question 45

Determine three positive numbers x, y, z that maximize under the condition x + y + z = 17.

Question 46

Consider the function Determine all local maxima, minima and saddle points of f.
Does f have a global maximum?

Question 47

Find the maximum and minimum values of the function subject to the constraint .

Question 48

Suppose the quantity, q, of a good produced depends on the number of workers, w, and the amount of capital, k, invested and is represented by the Cobb-Douglas function .In addition, labor costs are $20 per worker and capital costs are $20 per unit, and the budget is $3680.Using Lagrange multipliers, find the optimum number of units of capital.

Question 49

A company manufactures a product using x, y and z units of three different raw materials.The quantity produced is given by the function .Suppose the cost of the materials per unit is $20, $25 and $75 respectively.
Find the maximum production if the budget is limited to $6000.

Question 50

The Green Leaf Bakery makes two types of chocolate cakes, Delicious and Extra Delicious.Each Delicious requires 0.1 lb of European chocolate, while each Extra Delicious requires 0.2 lb.Currently there are only 233 lb of chocolate available each month.Suppose the profit function is given by: $p ( x , y ) = 151 x - 0.2 x ^ { 2 } + 200 y - 0.1 y ^ { 2 }$ where x is the number of Delicious cakes and y is the number of Extra Delicious cakes that the bakery produces each month.
(a)How many of each cake should the bakery produce each month to maximize profit?
(b)What is the value of $\lambda$ in part (a)(if $\nabla p = \lambda \nabla g$ )? What does it mean?
(c)It will cost $19.00 to get an extra pound of European chocolate.Should the bakery buy it?

Question 51

A coffee company sells three brands of coffee.Brand A costs p₁ dollars per can, brand B costs p₂ dollars per can, and brand C costs p₃ dollars per can.The demand (in hundreds of cans)depends on the prices as follows:
demand for brand A _,
demand for brand B _,
demand for brand C .
The company can produce 69,000 cans.What selling prices optimize the total revenue?

Question 52

Suppose that Find the minimum value of the function f when (x,y)is constrained to lie on or inside the triangle with vertices (0,-2), (0,1), and (1,-2).Give your answer to 4 decimal places.

Question 53

The following results are obtained when optimizing f(x, y)subject to the constraint g(x, y)= 35.The maximum value is f(5, 7)= 39, the Lagrange multiplier $\lambda$ = 4 (when $\nabla f = \lambda \nabla g$ )and $\nabla f = 2 \vec { i } - 5 \vec { j }$ .What is g(5, 7)?

Question 54

A company manufactures a product using x, y and z units of three different raw materials.The quantity produced is given by The production is described by the function .Suppose the cost of the materials per unit is $20, $15 and $24 respectively.
(a)Find the cheapest way to produce 6300 units of the product.
(b)Find the value of in and interpret this value.

Question 55

Use Lagrange multipliers to find the minimum value of 4xy on the circle .

Question 56

Let .The constraint g(x, y)= 3 is sketched in the picture below.
In the picture, locate the point where f will have a global maximum subject to the constraint g(x, y)= 3.

Question 57

The Lagrange multipliers needed to find the maximum and minimum values of 8xy on the circle is , where .Estimate the maximum and minimum values of 8xy subject to the constraint .

Question 58

Given that the quadratic Taylor polynomial of f at (4, 5)is decide whether is a critical point.If so, identify what sort of critical point it is.

Question 59

The following results are obtained when optimizing f(x, y)subject to the constraint g(x, y)= 39.The maximum value is f(5, 7)= 42, the Lagrange multiplier $\lambda$ = 3 (when $\nabla f = \lambda \nabla g$ )and $\nabla f = 3 \vec { i } - 3 \vec { j }$ .If the constraint condition is changed to g(x, y)= 40, what will be new maximum value of f(x, y)?

Question 60

Let be a vector in space with a, b > 0.
Compute the cross product and then use the result and the Lagrange Multiplier method to find the values of a and b such that the magnitude of the cross product is the largest with

Question 61

Let $f ( x , y ) = a x ^ { 2 } + b x - y ^ { 2 } + c y + d$ , for constants a, b, c, and d, with $a \neq 0$ .
The constants can be chosen in such a way that f will have a local minimum.

Question 62

Suppose there are two electric generators that burn natural gas and whose efficiency declines with output.The energy output is an increasing (but concave down)function of fuel input.Specifically, say
Output of generator 1 is

Question 63

Let $L ( x , y ) = a x + b y$ .
There exist values of a and b so that $L ( x , y )$ takes a minimum value of 8 on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$ at the point $\left( \frac { 1 } { 2 } , \frac { - \sqrt { 3 } } { 2 } \right)$ .

Question 64

An exam question asks students to find the maximum of on the circle , and the gradient vectors of f and g at that point.A student gave the following

Question 65

Find three positive numbers whose product is 11 and whose sum is a minimum.What is the minimum sum?

Question 66

Let .Let satisfy .Explain why the maximum of f subject to the constraint cannot occur at the point .

Question 67

The point (0,0)is a critical point for the function $f ( x , y ) = x ^ { 4 } + y ^ { 4 }$ .What kind of critical point is it?

A)local minimum
B)saddle point
C)local maximum
D)none of these

Question 68

Let .Find values of a and b so that takes a maximum value of 4 on the unit circle at the point .

Question 69

Let $f ( x , y ) = \sin x + \sin y + \cos ( x + y )$ in the square S bounded by $0 \leq x \leq \pi , 0 \leq y \leq \pi$ .Then $\left( \frac { \pi } { 6 } , \frac { \pi } { 6 } \right)$ is a critical point.What kind of critical point is it?

A)local maximum
B)local minimum
C)saddle point
D)none of these

Question 70

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Find the critical points of f.

Question 71

Let $f ( x , y ) = k x ^ { 3 } - 3 k x + y ^ { 2 }$ where k $\neq$ 0.Determine the values of k (if any)for which the critical point at (-1, 0)is a local minimum.

Question 72

Find three numbers x, y, and z, such that and is minimal.What is this minimal sum?

Question 73

It can be shown that $( 0,0 ) , ( 0,1 / 2 ) , ( 0 , - 1 / 2 ) , ( 1,0 ) , \text { and } ( - 1,0 )$ are the critical points of the function $f ( x , y ) = \left( x ^ { 2 } + y ^ { 2 } \right) e ^ { - x ^ { 2 } - 4 y ^ { 2 } }$ .Which of the following are classified correctly? Select all that apply.

A)(0,0)is a local minimum.
B)(1,0)is a local maximum.
C)(-1,0)is a local maximum.
D) $( 0,1 / 2 )$ is a saddle point.
E) $( 0 , - 1 / 2 )$ is a local minimum.

Question 74

The level curves of f(x, y)are shown in the picture below. (a)Estimate the global maximum and minimum of f on the closed triangular region D with vertices at (-1, -1), (2, -1)and (-1, 2).
(b)Find the critical point(s)of f in the interior of the region D.
(c)Find the critical point(s)of f along the boundary of D.

Question 75

Find the maximum and minimum values of subject to the constraint .

Question 76

The temperature at each point in the first quadrant is given by .Find the hottest point in the first quadrant and determine its temperature.

Question 77

Find the maximum and minimum values of subject to the constraint , with .Your answers may depend on c.