The required value of the given inverse function f⁻¹(22) will be 7.
What is an inverse function?The inverse function is defined as a function obtained by reversing the given function.
Given that f(7) = 22
To determine the value of the given inverse function f⁻¹(22)
An inverse function's main objective is to show the value that occurs in a specific function value. Which means :
If we have f(x) and f⁻¹(x), then
if f(a) = b then f⁻¹(b) = a
As per the given question, we have
⇒ f(7) = 22 then f⁻¹(22) = 7
Therefore, the required value of the given inverse function f⁻¹(22) will be 7.
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Find the slope of each line. Tell whether the slop is Positive, Negative, Undefined or Zero
a manufacturer of transistors claims that its transistors will last an average of 1000 hours. to maintain this average, 25 resistors are tested each month. you may assume that the distribution of the lifetime of a transistor is normal. what conclusions can you draw from a sample that has a mean of 1010 and a standard deviation of 60?
Do not reject H(0). There is not sufficient evidence to a manufacturer of transistors claims that its transistors will last an average of different than 1000 hours. at 0.01 significance level.
In the given question, a manufacturer of transistors claims that its transistors will last an average of 1000 hours.
To maintain this average, 25 resistors are tested each month.
We may assume that the distribution of the lifetime of a transistor is normal.
We have to calculate a sample that has a mean of 1010 and a standard deviation of 60.
So let us consider
X : Life time of a transistor
So we have
N=25
x=1010
s=60
Since sample size n(25≤30) is small and population standard is unknown so we use t statistic for hypothesis testing.
Now our null and alternate hypotheses are given as:
H(0): μ=1000
H(1): μ≠1000 (two tailed test)
Under Null hypothesis our test statistic is;
t=(x-μ)/(s/√n)
t=(1010-1000)/(60/√25)
t=0.83
Critical Value:
t(α,(n-1))=t(0.01, 24)
t(α,(n-1))=2.80
We have given level of significance α=0.01 and also we find the t-score 0.83 and degree of freedom=24
p value
The Excel formula is used to calculate the p value is:
P value =2×(1- (t(24) > 0.83))
P value =2×(1-0.7926)
P value =2×0.2074
P value =0.4147
Since p value=0.42 greater than α=0.01 means that deviation from the null hypothesis is not statistically significant, and the null hypothesis is not rejected and we claims that its transistors will last an average of 1000 hours.
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Which of the following is a equation of x-axis 1)y-0. 2)x-0. 3)x-0. 4)y-0
Answer:
none of the listed options
Step-by-step explanation:
the equation for x axis is y=0
Please help me im so close to getting caught up but I’m still behind and I only have 2 hours!!
The ratios of calories Robyn burned on Thursday to the number of calories she burned for the week is 3 : 20.
What is ratio?Ratio, in math, is a term that is used to compare two or more numbers. In other words, a ratio is a comparison of two or more numbers that indicates their sizes in relation to each other.
Therefore, the ratios of calories Robyn burned on Thursday to the number of calories she burned for the week can be calculated as follows:
Let's represent it in fractions,
calories burn on Thursday / calories burn for the week
calories burn on Thursday = 360
calories burn for the week = 488 + 551 + 610 + 360 + 391 = 2400
Hence,
360 / 2400 = 36 / 240 = 12 / 80 = 3 / 20
Therefore, the ratio is 3:20
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21) Submarine A is sent to explore the ocean floor in the Atlantic Ocean. It descends to 126
feet below sea level, For the next 4 ½ hours, it descends further at a rate of 15 feet per hour
and then reaches its destination.
Submarine B is sent to explore the ocean floor in the Pacific Ocean. It begins its descent at
sea level and reaches the same elevation as Submarine A over the course of 6.5 hours. At
what rate does Submarine B descend? Show and explain your thinking.
The rate at which Submarine B descends is 29.8 feet per hour.
What is the rate?The rate refers to the average speed of an object in motion from one point to another.
The rate can be determined by dividing the distance by the time.
The rate or average speed is the quotient between distance and time.
The distance descended by Submarine A = 193.5 feet (126 + 4 ½ x 15)
The time Submarine B descends = 6.5 hours
Submarine B's rate of descent = 29.77 feet (193.5/6.5) per hour
= 29.8 feet
Thus, we can conclude that Submarine B descends at the rate of 29.8 feet every hour, whereas, Submarine A descends 15 feet per hour.
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b.
5-50.
Use your new 30°-60°-90° and 45°-45°-90° triangle patterns to quickly find the lengths of
the missing sides
in
each of the triangles below. Do not use a calculator. Leave answers in exact form. Note: The triangles are not
necessarily drawn to scale.
a.
41
10
45°
8
4
30°
4
16
8
0
c.
d.
854
45°
9
30°
(2)
مع
60°
f.
60⁰
6
6
45°
The length of the sides of the special triangles, found using the relationship between the sides of the triangles are as follows;
(1) Length of the legs is 20 each
Length of the hypotenuse = 20·√2
(2) Length of the leg = 5
Length of the hypotenuse = 5·√2
5-50 a. Length of the leg is 10·√3
Length of the hypotenuse is 20
b. Length of the leg = 4
Length of the hypotenuse = 4·√2
c. Length of each leg = 8
d. Length of the leg = 9·√3
Length of the hypotenuse = 18
e. Length of the shorter leg = 4.5
Length of the longer leg = 4.5·√3
f. Length of the legs = 3·√2 each
5-51
Special triangles are the 30°, 60°, 90°, triangle, and the 45°, 45°, 90°, triangles.Please find attached the diagrams of special triangles, created with MS Word
What are triangles?
A triangle is a three sided polygon, that has a sum of interior angles of 180°
The lengths of the missing sides in the triangles found using the 30°–60°–90°, and 45°–45°–90°, triangle patterns are as follows;
In a 30°, 60°, 90° triangle, if the length of the shortest leg is x, the length of the hypotenuse is 2·x, and the length of the other leg is x·√3
In a 45°, 45°, 90° triangle, the length of the legs are the same, and the length of the hypotenuse is √2 multiplied by the length of one leg.
(1) The length of the legs are the same and equivalent to 20
The length of the hypotenuse side is 20·√2
(2) The length of the legs are the same and equivalent to 5
The length of the hypotenuse 5·√2
5-50 a. In the 30°, 60°, 90° triangle, the length of the adjacent, longer leg is 10·√3
The length of the hypotenuse side is 2·x
b. The triangle is a 45°, 45°, 90°, triangle
The length of the legs are 4 unit each
The length of the hypotenuse is 4·√2
c. The length of the hypotenuse in the 45°, 45°, 90° is 8·√2, therefore, the length of the legs are;
8 units each
d. The length of the shortest leg in the 30°, 60°, 90°, right triangle is 9, therefore, the length of the hypotenuse side is 2 × 9 = 18
The length of the longest leg is 9·√3
e. The length of the hypotenuse in the 30°, 60°, 90°, right triangle is 9 units, therefore, the length of the shortest leg is 4.5 units
The length of the longest leg is 4.5·√3
f. The length of the hypotenuse side of the 45°, 45°, 90°, right triangle is 6, therefore;
The length of the legs are 6/√2 = 3·√2
5-51
The dimensions of the side length of the special right triangles are;
In a 30°-60°-90° triangle, if the length of the shortest leg is x, then;
The length of the hypotenuse is 2·x
The length of the longer leg is x·√3
In a 45°-45°-90° right triangle, the length of the legs are the same, and the length of the hypotenuse is √2 multiplied by the length of one leg.
Please attached diagrams of the 45°, 45°, 90°, and the 30°, 60°, 90° triangles created with MS Word
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A recipe for pizza sauce calls for 11 cans of tomatoes and 44 teaspoons of oregano. How many teaspoons of oregano would you need for a single can of tomatoes?
Answer:
One can of tomatoes needs 4 teaspoons of oregano.
Step-by-step explanation:
[tex]\frac{11}{44}[/tex] = [tex]\frac{1}{x}[/tex] Cross multiply and solve for x
11x = 44 Divide both sides by 11
x = 4
One can of tomatoes needs 4 teaspoons of oregano.
(6.8 times 10 to the power 2) times(1.3 times 10 to the power -3)
The answer of the given scientific multiplication is, 8.84 x 10^-1.
What is scientific multiplication?
Multiply the decimal numbers to multiply the numbers in scientific notation. Add the 10 power exponents after that. In scientific notation, place the new power of 10 with the decimal.
Given:
(6.8 x 10^2) x (1.3 x 10^-3)
We have to solve this using scientific multiplication.
First to multiply the decimals,
6.8 x 1.3 = 8.84
Now to solve
10^2 x 10^-3 = 10^(2-3) = 10^-1
⇒ (6.8 x 10^2) x (1.3 x 10^-3) = 8.84 x 10^-1
Hence, the answer of the given scientific multiplication is, 8.84 x 10^-1.
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Which scatter plot shows the best line of fit?
Multiple choice question.
A)
B)
C)
D)
Answer:
D
Step-by-step explanation:
The line of best fit is a line that represents the “average” between all of the points. It’s typically in the middle of all of the points. In this case, I’d go with D because 4 points lie on the line of best fit and the line trends sharply upwards, following the general direction of the points.
How do I find a perpendicular line to a vertical line graph eg: x=2?
Answer:
Any line that is in the form
y =
Step-by-step explanation:
See the graph below. Perpendicular means that the lines meet at 90°. You will see the black line graphed that is x = 2. Then you will see two graphs in red. y = 6 and y = -1. any line in the form y = some number will be perpendicular to x = 2.
simplify the expression: (3x10^8)(6x10^-4)
The answer is 18x10^4 using simple arithmetic calculations and exponential function.
What are arithmetic calculations?
Arithmetic calculations are mathematical operations that involve the addition, subtraction, multiplication, and division of real numbers. Arithmetic is used to solve problems and execute tasks in everyday life, and is an integral part of mathematics. It is also used to analyze data and solve equations. Arithmetic involves the use of numbers, symbols, and operations in order to solve problems. Examples of arithmetic operations include addition, subtraction, multiplication, division, and exponentiation. The order of operations is an important part of arithmetic, as it dictates which operations take precedence over others. Arithmetic is used in a variety of applications, such as in engineering, finance, education, and science. It can also be used to solve puzzles and games. Arithmetic calculations can be used to help people better understand the world around them, as well as to help them make better decisions in life.
Two exponential function having same base , then powers can be added or subtracted.
In given equation, base is of 10 and powers are 8 and -4.
now the power will be 4
and 3 and 6 get multiplied i.e. 18.
So, the answer is 18x10^4.
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a person leaves home and walks 4 miles west, then 5 miles southwest. how far from home is she? miles in what direction must she walk to head directly home? degrees north of east
The person is 8.325 miles far from her home and she should walk in the direction of 64.86° North Of East to head directly home .
In the question ,
from the figure ,
we see that point O is home of the person ,
it is given that person leaves the home and walks 4 miles west to point A , then 5 miles in south west direction to the point B .
So ,distance OA = 4 miles
AB = 5 miles .
In triangle OAB ,
angle OAB = 45+90 = 135°
Using Cosine Rule ,
OB² = OA² + AB² - 2*OA*AB*Cos(135)°
= 16 + 25 + 28.2843 = 69.2843
So ,OB = √69.2843 = 8.325 miles .
Now using Sine Rule ,
Sinθ/OA = Sin(135°)/OB
Sinθ = 4*Sin(135°)/8.324
Sinθ = 0.339792
So , θ = 19.86°
hence the direction from North of East is
= 45° + θ
= 45° + 19.86°
= 64.86° North Of East .
Therefore , the distance is 8.325 miles and the direction is 64.86° North Of East .
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a biologist wants to calculate the number of fish in a lake. on may 1 she catches a random sample of 60 fish, tags them, and releases them. on september 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. to calculate the number of fish in the lake on may 1, she assumes that 25% of these fish are no longer in the lake on september 1 (because of death and emigrations), that 40% of the fish were not in the lake may 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the september 1 sample are representative of the total population. what does the biologist calculate for the number of fish in the lake on may 1?
The total number of fishes present in the lake on my 1 is 840.
Explain the term percentage of the number?A figure or ratio that may be stated as a percentage of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100.The given data for the question-
42 fish were present in May out of the 70 fish that were caught in September, 40% of which were absent in May. Noting that the 25% death rate has no impact on the answer since both tagged and untagged fish die, the percentage the tagged fish inSeptember is equivalent to a percentage of tagged fish in May.
3/42 = 60/x
x = 60 *42 / 3
x = 840
Thus, the total number of fishes present in the lake on my 1 is 840.
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Which of the following expressions is equal to
(sin 60°)(cos 30º) + (cos 60°)(sin 30°) ?
F. cos(60° -30°)
G. cos(60° +30°)
H. sin(60° -30°)
J. sin(60° +30°)
K. sin(60* 430")
+ °
2
The answer is cos 60° × cos 30° + sin 60° × sin 30° = √3/2.
Find the solution ?The fundamental guidelines and procedures to simplify any algebraic statement are as follows:By multiplying factors, remove any grouping symbols like brackets and parenthesis.
Simplify each term.
34+14
Combine fractions.
Combine the numerators over the common denominator.
3+14
Simplify the expression.
Add 3
and 1
44
Divide 4by 4
= 1.
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Max Says that the product 30 x 10 by 4 has exactly four zeros
Max Says that the product 30 x 10 by 4 has exactly four zeros is not true.
What is Multiplication?Multiplication is a method of finding the product of two or more numbers
We need to check whether product 30 x 10 by 4 has exactly four zeros
firstly we have to multiply thirty with ten
Now multiply 30×10=300
Which has two zeros and now we have to divide it by 4
300/4
Three hundred by four
75
Seventy five.
Hence, Max Says that the product 30 x 10 by 4 has exactly four zeros is not true.
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Which of the following statement is true ? a) X^2 + 5x -3 is a linear polynomial b) X^2+4x-1 is a binomial c) (x+1) is a monomial d) 5x^3 is a monomial
Answer:
d. [tex]5x^3[/tex] is a monomial
Step-by-step explanation:
⭐What is a monomial?
as "mono" means 1, monomial means an expression or equation that consists of 1 term onlyD is correct because the expression [tex]5x^3[/tex] consists of 1 term only.
⭐What is a linear polynomial?
a linear polynomial is an expression or equation that has a degree of 1⭐What is the degree of a polynomial?
the degree of a polynomial is the largest exponent found in the polynomialA is incorrect because the largest exponent in the expression [tex]x^2 + 5x -3[/tex] is 2, which makes expression A a quadratic polynomial.
⭐What is a binomial?
as "bi" means 2, a binomial is an expression or equation that consists of 2 terms onlyB is incorrect because the expression [tex]x^2 + 4x - 1[/tex] consists of 3 terms, not 2 terms. B is a trinomial.
C is incorrect because the expression [tex]x+1[/tex] consists of 2 terms, not 1 term. C is a binomial.
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Magan went to the grocery store and purchased cans of soup and frozen dinners. Each can of soup has 350 mg of sodium and each frozen dinner has 650 mg of sodium. Magan purchased a total of 11 cans of soup and frozen dinners which collectively contain 4750 mg of sodium. Determine the number of cans of soup purchased and the number of frozen dinners purchased.
and please add how many cans of soup and boxed dinners she purchased
The number of cans of soup bought is 8 and the number of frozen dinners purchased is 3.
How many cans of soup and frozen dinners was purchased?s + f = 11 equation 1
350s + 650f = 4750 equation 2
Where:
s = number of soup cans that Magan bought f = number of frozen dinner that Magan boughtThe elimination method would be used to solve for the required values:
Multiply equation 1 by 350
350s + 350f = 3850 equation 3
Subtract equation 2 from equation 3 :
300f = 900
f = 900 / 300
f = 3
Substitute for f in equation 1 :
s + 3 = 11
s = 11 - 3
s = 8
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Someone help please thank you show your work 100 points!!
Answer:
A
Step-by-step explanation:
y=-2 when x=-4 y=-4 when x=-8
what's given: 4. what's given:2
-4k=-2. -8k=-4
-4/4. -2/4. -8/8. -4/8
k=0.5. k =0.5
0.5(2) =1. 0.5(2)=1
Answer:
(a) no; (b) no; (c) yesy=5x; b=1/2a; m = 5nk = 5; x = 9.8Step-by-step explanation:
You want to identify the proportional relations in the given tables and equations, and you want to find and use the constant of variation given that y=35 when x=7.
Proportional relationA proportional relation is one that has an equation of the form ...
y = kx
One variable is a constant multiple of the other. If the relation cannot be put in this form, it is not a proportional relation.
ApplicationThe value of k in the above equation can be found as ...
k = y/x . . . . . . . divide both sides by x
This is a constant if the relation is proportional.
1a.y/x = -8/-4 = -4/-2 ≠ 18/6 . . . . . not a proportional relation
1b.y/x = -4/-5 ≠ 10/2 . . . . . not a proportional relation
1c.y/x = -12/-3 = 24/6 = 32/8 = 4 . . . . . a proportional relation
2.
You are looking for equations of the form y = kx. The only ones having that form are ...
y = 5xb = 1/2am = 5nThe others represent inverse proportions and a linear (not proportional) relation.
3.As we saw above, the constant of variation is ...
k = y/x = 35/7
k = 5
Then the value of x can be found from ...
49 = 5x
49/5 = x = 9.8
Sophia earns $16 per hour and worked 7.5 hours per day. She spent
$18 per workday on lunch. Sophia saves a quarter of her remaining
money in her savings per week.
A.
Write an expression representing the amount of money
Sophia put in her savings account at the end of the week.
B. How much money did she put in her savings?
The answers to both subparts are shown:
(A) The expression is: 4x = 714
(B) Money she put in her savings account each week: $178.5
What is an expression?An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.
Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression.
In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
Let's assume she works 7 days a week.
Then 7.5 * 16 = $120 (1-day earning)
30 days: 120 * 30 = 840
$18 each day on lunch: 18 * 7 = $126
Remaining: 840 - 126 = $714
A quarter of $714: 714/4 = $178.5
(A) The expression will be:
4x = 714
(B) Money she put in her savings account each week:
4x = 714
x = 714/x
x = $178.5
Therefore, the answers to both subparts are shown:
(A) The expression is: 4x = 714
(B) Money she put in her savings account each week: $178.5
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Which of the following is a linear polynomial?
(A) x +5
(B) x2 +5
(C) x3 +5
(D) x4 +5
the correct answer is A cause × +5 is linear polynomial
when physicians diagnose arterial blockages, they quote the reduction in flow rate. if the flow rate in an artery has been reduced to 10.0% of its normal value by a blood clot and the average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?
The radius reduced to 57.3% of its normal value.
The expression of Poiseuille's law for flow in a tube is:
Q = Δpπr⁴/8ηl
where:
Q = is the flow rate
P₁ = is the pressure at first end of vain
P₂ = is the pressure at the other end
η = is the viscosity of the concrete
r - radius of pipe
l = is the length
Re arrange the expression for the radius of the pipe,
r = (8Qηl/πΔP)¹/⁴
the flow rate in an artery has been reduced to 10.0%and the difference in blood pressure is increased by 20%.
so,
Q' = 0.100Q and
ΔP' = 1.20ΔP
SUbstitute the above points.
r' = (8( 0.100Qηl/π(1.20ΔP))¹/⁴
= 0.537(8Qηl/πΔP)¹/⁴
Combine the expression of radius before the flow and pressure changes and the expression derived or the new radius,
r' = 0.537r
Hence the radius reduced to 57.3% of its normal value.
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true or false considered multiple regression which is a sample at 36 observations for a given eggs and fitted why the 95% approximate prediction interval for individual why is wider than the corresponding approximate 95% confidence interval for the conditional mean a y
The statement "considered multiple regression which is a sample at 36 observations for a given eggs and fitted why the 95% approximate prediction interval for individual why is wider than the corresponding approximate 95% confidence interval for the conditional mean" is false.
Here we have given the statement and we need to identify whether this one is true or not.
As we all know that then Confidence interval is narrower than the prediction interval because Prediction intervals must account for both the uncertainty in estimating the population mean, plus the random variation of the individual values.
Therefore the prediction interval is always wider than a confidence interval.
And also here the prediction interval will not converge to a single value as the sample size increases.
Therefore, the statement is incorrect.
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(2x^2 - 4x + 3) + (5x - 6x^2 - 1)
Answer:
The answer is (-4x² + x + 2)
Step-by-step explanation:
(2x² - 4x + 3) + (5x - 6x² - 1)
open the brackets
2x² - 4x + 3 + 5x - 6x² - 1
rearrange the priority order
2x² - 6x² - 4x + 5x + 3 - 1
combine like terms
-4x² + x + 2
Thus, The answer is (-4x² + x + 2)
last year, 54% of business owners gave a holiday gift to their employees. a survey of business owners conducted this year indicates that 32 of them plan to provide a holiday gift to their employees. suppose the survey results are based on a sample of 80 business owners. (a) suppose the business owners in the sample did as they plan. compute the p-value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year. find the value of the test statistic. (round your answer to two decimal places.)
The p-value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year is 0.0060 and the value of test statistic is - 2.51
54% of business owners gave a holiday gift to their employees. a survey of business owners conducted this year indicates that 32 of them plan to provide a holiday gift to their employees.
The survey results are based on a sample of 80 business owners
We need to find the value of test statistic.
Expected number = np = 32
80 x p = 32
p = 0.4
This is a left tailed test
test statistic is (p^ - p)/√(pq/n)
= (0.40 - 0.54)/√(.54 - (1 - 0.54) / 80)
= 0.14/√((0.54 - 0.46)/80)
= -2.51
p value = P(z < - 2.51)
= 0.00060366
= 0. 0060
Therefore, the p-value for a hypothesis test that can be used to determine if the proportion of business owners providing holiday gifts has decreased from last year is 0.0060 and the value of test statistic is - 2.51
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6. In a company, the number of men is 3.5 times more than the number of women. How many of the company's 108 employees are men?
Answer:
84
Step-by-Step Explanation:
A wire is used to anchor a 20-foot-tall pole. One end of the wire
is attached to the top of the pole. The other end is fastened to a
stake five feet away from the bottom of the pole. Find the length
of the wire.
The length of the wire attached to the pole is 20.62 feet
What is the length of the wire?The pole is perpendicular to the ground in the direction of the point where the wire is attached to the ground to form a right angle triangle.
A leg of the triangle is 20 feet(length of the pole from the surface of the ground to the top)
The other leg is5 feet(distance from the base of the pole to the point where the wire is attached to the ground)
The third side of the right angle triangle is the hypotenuse(length of the wire)
Applying Pythagoras theorem;
Hypotenuse² = Opposite² + Adjacent²
= 20 feet ² + 5 feet²
= 400 + 25
Hypotenuse² = 425
Find the square root of both sides
Hypotenuse = √425
= 20.6155
Approximately,
Hypotenuse = 20.62 feet
In conclusion, the hypotenuse side of the right angle triangle is 20.62 feet
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a manufactured lot of buggy whips has 20 items, of which 5 are defective. a random sample of 5 items is chosen to be inspected. find the probability that the same contains exactly one defective item
This indicates that the probability is 0.4402% without replacement.
What is Probability?The ratio of good outcomes to all possible outcomes of an event is known as the probability. The number of positive results for an experiment with 'n' outcomes can be represented by the symbol x.
The probability of an event can be calculated using the following formula.
Probability(Event) = Positive Results/Total Results = x/n
In order to better grasp probability, let's look at a straightforward application.
Let's say we need to forecast if it will rain or not. Either "Yes" or "No" is the appropriate response to this query.
There is a chance that it will rain or not. Here, probability can be used. Using probability, one may forecast the results of a coin toss, a roll of the dice, or a card draw.
According to our question-
n = 5
x = 1
p = 5/20 = 0.25
15C4 = 15! / 4! * 11! = 1365
P = (5C1 * 15C4) / 20C5
Replacing:
P = 5 * 1365/15504
P = 0.4402
Hence, This indicates that the probability is 0.4402% without replacement.
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Factor the polynomial
below completely:
14a-7
(Show work please)
lagrange multiplier what are the points on the intersection of the ellipsoid with the plane that are respectively closest and furthest from the origin?
The points on the intersection of the ellipsoid with the plane that are respectively closest and furthest from the origin are
(2–√,−2–√,2−22–√)
(−2–√,2–√,2+22–√)
Using Lagrange multipliers we attempt to find the extrema of f(x,y,z)=x2+y2+z2 given that g(x,y,z)=x−y+z−2=0 and that h(x,y,z)=x2+y2−4=0.
Given,
∇f=⟨2x,2y,2z⟩
∇g=⟨1,−1,1⟩
∇h=⟨2x,2y,0⟩
Extrema satisfy the condition that ∇f=μ∇g+λ∇h for some λ,μ∈R.
This is to say,
2x=2λx+μ
2y=2λy−μ
2z=μ
If λ=1 then μ=0 and so z=0, the g constraint tells us that x=y+2, and the h constraint tells us that y2+(y+2)2=4, meaning that either y=0 or y=2. This provides us with two crucial points in addition to the g constraint:
(2,0,0)
(0,−2,0)
Now assume λ≠1, and so
x=μ2−2λ
y=−μ2−2λ=−x
Since x=−y, we have that x=±2–√, y=∓2–√. Using the g constraint, our two critical points are
(2–√,−2–√,2−22–√)
(−2–√,2–√,2+22–√)
And then it's east to determine which is the max and which is the min out of these four critical points.
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