What value of x is in the solution set of 2x-3> 11 - 5x?
-3
2

Answer:

wow-

Step-by-step explanation:

ngl, thats hard.

Answer:

14

Step-by-step explanation:

8/9 = 0.888888888......

<5 round down

\(\geq\)5 round up

8/9 = 0.888888888....

larger than 0.5

the answer is 14

Answer:

1. B. The bag of candy produced last Monday is different from the standard bag.

2. A. The bag of candy produced last Monday are like the standard bag.

3. B. The bag of candy produced last Monday is different from the standard bag.

Step-by-step explanation:

Given - A company that makes hard candy have a standard bag of hard candy with 150 pieces. The hard candy has three distinct colors red, white and orange and equal proportion of each candy is present in the standard bag. The manager wants to know whether the bags produced last Monday were similar to the standard bag. To test this, they plan to choose a random bag from the batch and compare it with the standard bag. You will have to answer questions below to assist the manager in comparing the two bags.

To find - 1. What is the alternate hypothesis for this exercise?

              2. What is the null hypothesis for this exercise?

              3. After performing the necessary calculations, the chi-square value of the test is 6.7. The cut-off/critical value of chi-square test is 5.991 (for 5% chance).  What is your conclusion about the bags based on this result?

Proof -

1.

The alternate hypothesis is - B. The bag of candy produced last Monday is different from the standard bag.

2.

The null hypothesis is - A. The bag of candy produced last Monday are like the standard bag.

3.

The conclusion is - B. The bag of candy produced last Monday is different from the standard bag.

6.1). the rate of depreciation, r, is approximately 10.77%.

6.2.1). Ncominkosi's monthly installment amount was approximately R 10,505.29.

6.2.2).  Ncominkosi borrowed approximately R 377,510.83 from the bank.

6.1) Let's assume the original price of the laptop is P. According to the reducing balance method, the value of the laptop after 4 years can be calculated as P * (1 - r/100)^4. We are given that this value is 31% of the original price, so we can write the equation as P * (1 - r/100)^4 = 0.31P.

Simplifying the equation, we get (1 - r/100)^4 = 0.31. Taking the fourth root on both sides, we have 1 - r/100 = ∛0.31.

Solving for r, we find r/100 = 1 - ∛0.31. Multiplying both sides by 100, we get r = 100 - 100∛0.31.

Therefore, the rate of depreciation, r, is approximately 10.77%.

6.2.1) To determine the monthly installment amount, we can use the formula for calculating the monthly payment on a loan with compound interest. The formula is as follows:

\(P = \frac{r(PV)}{1-(1+r)^{-n}}\)

Where:

P = Monthly payment

PV = Loan principal amount

r = Monthly interest rate

n = Total number of monthly payments

Let's calculate the monthly installment amount for Ncominkosi's loan:

Loan amount = Total amount paid to the bank - Interest

Loan amount = R 596,458.10 - R 0 (No interest is deducted from the total paid amount since it is the total amount paid)

Monthly interest rate = Annual interest rate / 12

Monthly interest rate = 9.5% / 12 = 0.0079167 (rounded to 7 decimal places)

Number of monthly payments = 6 years * 12 months/year = 72 months

Using the formula mentioned above:

\(P = \frac{0.0079167(Loan Amount}{1-(1+0.0079167)^{-72}}\)

Substituting the values:

\(P = \frac{0.0079167(596458.10}{1-(1+0.0079167)^{-72}}\)

Calculating the value:

P≈R10,505.29

Therefore, Ncominkosi's monthly installment amount was approximately R 10,505.29.

6.2.2) To determine the amount of money Ncominkosi borrowed from the bank, we can subtract the interest from the total amount he paid to the bank.

Total amount paid to the bank: R 596,458.10

Since the total amount paid includes both the loan principal and the interest, and we need to find the loan principal amount, we can subtract the interest from the total amount.

Since the interest rate is compounded monthly, we can use the compound interest formula to calculate the interest:

\(A=P(1+r/n)(n*t)\)

Where:

A = Total amount paid

P = Loan principal amount

r = Annual interest rate

n = Number of compounding periods per year

t = Number of years

We can rearrange the formula to solve for the loan principal:

\(P=\frac{A}{(1+r/n)(n*t)}\)

Substituting the values:

Loan principal (P) = \(\frac{596458.10}{(1+0.095/12)(12*6)}\)

Calculating the value:

Loan principal (P) ≈ R 377,510.83

Therefore, Ncominkosi borrowed approximately R 377,510.83 from the bank.

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confidence interval is (53.0, 60.8), where 53.0 is the lower limit and 60.8 is the upper limit. This means we are 95% confident that the population means lies within this interval.

a) The length of a confidence interval is twice the margin of error. In this case, the margin of error is 3.9, so the length of the confidence interval would be 2 * 3.9 = 7.8.

b) To obtain the confidence interval, we need the sample mean and the margin of error. Given that the sample mean is 56.9, we can construct the confidence interval as follows:

Lower limit = Sample mean - Margin of error = 56.9 - 3.9 = 53.0

Upper limit = Sample mean + Margin of error = 56.9 + 3.9 = 60.8

Therefore, the confidence interval is (53.0, 60.8), where 53.0 is the lower limit and 60.8 is the upper limit. This means we are 95% confident that the population means lies within this interval.

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a) The probability that the murder was committed with a knife: 0.55

b) The probability that the cook did it, given that the murder was committed with a knife: 0.275

To determine the probability that the murder was committed with a knife, we first need to find the total number of weapons.

The butler has four poison tipped pens, two crowbars and four knives and the cook has three rolling pins and seven knives.

So, the total knives: 4 + 7 = 11

crowbars: 2

poison tipped pens: 4

and rolling pins: 3

So, the total number of weapons: 11 + 2 + 4+ 3 = 20

The probability that the murder was committed with a knife:

p = 11/20

p = 0.55

Now we need to find the probability that the cook did it, given that the murder was committed with a knife.

P = 0.55 × 0.5

P = 0.275

The required probability is 0.275

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Answer:

-1

Step-by-step explanation:

\(i^1\cdot i^2\cdot i^3\cdot i^4= \\\\i^{1+2+3+4}= \\\\i^{10}= \\\\-1\)

Hope this helps!

Answer:

Step-by-step explanation:

Dont do it. Just take the detention

The P-value for the statistic falls between 0.001 and 0.01 (or 0.1%) since it is less than the typical alpha level of 0.05 used to reject or fail to reject the null hypothesis.

To find the p-value for the given t-statistic, we need to use a t-distribution table or calculator with 24 degrees of freedom.

Using a t-distribution table, we can find the p-value for a one-tailed test with 24 degrees of freedom as follows:

Look up the absolute value of the t-statistic in the table (2.63 in this case).

Find the row corresponding to the degrees of freedom (24 in this case).

The intersection of the row and column corresponding to the t-statistic will give us the p-value for the test.

Using this method, we find that the p-value for a one-tailed t-test with 24 degrees of freedom and a t-statistic of -2.63 is approximately 0.008.

Therefore, the P-value for the statistic falls between 0.001 and 0.01 (or 0.1%) since it is less than the typical alpha level of 0.05 used to reject or fail to reject the null hypothesis.

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There exists a unique linear operator ¯T : V/W → V/W such that ¯T ◦proj = proj ◦T.

To show the existence and uniqueness of the linear operator ¯T : V/W → V/W, we need to demonstrate that it satisfies the composition property ¯T ◦proj = proj ◦T.

First, let's consider the composition ¯T ◦proj. Given an element [v]W in V/W, where v is an element of V, the composition ¯T ◦proj maps [v]W to ¯T(proj([v])) in V/W. Since proj([v]) is the equivalence class of v modulo W, ¯T(proj([v])) is the equivalence class of T(v) modulo W.

Now, let's consider the composition proj ◦T. For any vector v in V, proj(T(v)) is the equivalence class of T(v) modulo W.

To show the existence and uniqueness of ¯T, we need to demonstrate that ¯T(proj([v])) = proj(T(v)) for all elements [v]W in V/W. This can be done by showing that the two compositions ¯T ◦proj and proj ◦T give the same result for any element v in V.

Once we establish the existence and uniqueness of ¯T, we can conclude that there exists a unique linear operator ¯T : V/W → V/W that satisfies ¯T ◦proj = proj ◦T.

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The statement a base class cannot contain a pointer to one of its derived classes is false because a base class can indeed contain a pointer to one of its derived classes.

In object-oriented programming, a base class can have a pointer to one of its derived classes. This is known as upcasting or polymorphism. Upcasting allows for the flexibility of treating derived class objects as instances of the base class.

By using pointers, a base class can refer to derived class objects and access their member functions and variables. This enables the base class to work with different derived classes without needing to know their specific types.

Pointers to derived classes can be stored in base class member variables or passed as function parameters. This allows for dynamic binding and the ability to invoke overridden functions based on the actual derived class type at runtime.

This concept is fundamental to achieving polymorphism and code reusability in object-oriented programming languages like C++ and Java. It facilitates the implementation of inheritance hierarchies and the ability to work with objects of different derived classes through a common base class interface.

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Answer:

B. 183

Step-by-step explanation:

m<L = (1/2)[m(arc)KN - m(arc)KM)]

35 = (1/2)(177x - 107x)

70 = 70x

x = 1

m(arc)KN = 177x = 177

m(arc)NMK = 360 - m(arc)KN

m(arc)MNK = 360 - 177

m(arc)MNK = 183

\((\stackrel{x_1}{-9}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{-17}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{-17}-\underset{x_1}{(-9)}}} \implies \cfrac{4 }{-17 +9} \implies \cfrac{ 4 }{ -8 } \implies - \cfrac{1 }{ 2 }\)

\(\bold{\huge{\blue{\underline{ Solution }}}}\)

Ellis bought total 24 soda cans but in packs

Each pack contain 6 cans

Therefore,

Total packs bought by Ellis

\(\sf{ = }{\sf{\dfrac{ 24}{6}}}\)

\(\sf{ = 4 \: packs }\)

Ellis bought 4 packs of soda cans

So,

Total cost paid by the Ellis

\(\sf{ = 4 {\times}{\pounds} 1.80 }\)

\(\sf{ = {\pounds}7.20 }\)

Thus, The total cost paid by Ellis is £7.2

Connor also bought 24 cans of soda but in packs

Each pack contain 8 soda cans

Total packs bought by Connor

\(\sf{ = }{\sf{\dfrac{ 24}{8}}}\)

\(\sf{ = 3 \: packs }\)

Connor bought 3 packs of soda cans

So,

Total cost paid by the Connor

\(\sf{ = 3 {\times}{\pounds}2.10 }\)

\(\sf{ = {\pounds}6.30 }\)

Thus, The total cost paid by connor is £6.30

From Above we can conclude that,

That is,

Ellis had to pay more than connor by

\(\sf{ = 7.20 - 6.30 }\)

\(\sf{ = {\pounds}0.90}\)

Hence, Ellis paid more than connor by £0.90.

Using proportions and the information given, it is found that:

-------------------------

\(W = \frac{134 - 19}{8} = 14.375\)

-------------------------

The intervals will be of:

19 - 33.375

33.375 - 47.750

47.750 - 62.125

62.125 - 76.500

76.500 - 90.875

90.875 - 105.250

105.250 - 119.625

119.625 - 134.

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Answer:

ly + 2 x − 3y

The angles for the complementary angles are 42° and 48°.

Complementary angles are the angles that have a value equal to 90°.

Since one angle is 6 degress less than the other angle. This will be x and x + 6.

Therefore, we'll equate them to 90°.

x + x + 6 = 90°

2x + 6 = 90°

2x = 90 - 6

2x = 84

Divide

x = 84 / 2

x = 42

x + 6 = 42 + 6 = 48

The angle are 42 and 48°.

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Answer:

Female: 93/328

Adult: 103/328

Baby: 61/328

Step-by-step explanation:

71 + 93 + 103 + 61 = 328

Male: 71/328

Female: 93/328

Adult: 103/328

Baby: 61/328

Answer:

$26.99

Step-by-step explanation:

17.50 original cost plus a 54.2% increase

First, divide percentage by 100 to get a decimal to divide by 54.2/100=0.542

Then multiply the new decimal by the original cost to find the amount marked up. 17.50 x 0.542= 9.485

Now add 9.485 to the original cost

17.50+9.485=26.985-> 26.99 rounded for the final answer

Answer : The cardinality of is 15.

It is the union of and and its cardinality is the sum of the cardinalities of the two sets. Thus, the cardinality of is 6 + 9 = 15. The cardinality of a set is the number of elements in the set. In other words, it is the count of the members of a set. For example, if you have a set containing the numbers 1, 2, 3, 4, and 5, then the cardinality of that set is 5, since there are five elements in the set.

In the given problem, is a finite set of cardinality 6. This means that has 6 elements. Similarly, has a cardinality of 9, which means that it has 9 elements. Since is the union of and , it will have all the elements of both and , which means it will have 6 + 9 = 15 elements. Therefore, the cardinality of is 15.

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Answer:

Side 3 = 5cm

Step-by-step explanation:

135/9=15

15/3=5

Answer:

n=11

The first step is to use distributive property on the right side of the equation. Hope that helps

Option C best describes the limiting values of T and a under the conditions given. In this case, T represents tension and a represents acceleration.

Without the specific conditions mentioned, it is impossible to determine the exact limiting values of T and a. However, certain options can be ruled out based on common sense and physical laws. For example, option C (T=mg and a=0) is not possible as the tension in a string cannot be equal to the weight of an object. Option E (T=0 and a=[infinity]) is also not possible as a mass cannot have zero tension and infinite acceleration.

Based on these eliminations, the most reasonable options are A (T=0 and a=0) and D (T=[infinity] and a=g). In the former case, the object is not moving and there is no tension in the string. In the latter case, the object is in free fall and the tension in the string is negligible compared to the weight of the object.

However, it is important to note that the exact limiting values of T and a will depend on the specific conditions of the scenario, such as the mass of the object and the angle of the string.

Option C best describes the limiting values of T and a under the conditions given. In this case, T represents tension and a represents acceleration. When T=mg and a=0, it means that the tension in the system is equal to the gravitational force acting on the mass (mg) and the system is in equilibrium with no acceleration. This is a common scenario when an object is hanging from a rope or cable and not moving. The other options do not represent stable or realistic conditions for a physical system.

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Three balls are drawn simultaneously from an urn. There are 24 ways to draw one ball of each color, 136 ways to draw at least one green ball and there are 56 + 1 + 1 = 58 ways to draw the same color ball.

Probability is a fascinating branch of mathematics that deals with the study of random events. It is essential in understanding the likelihood of certain events occurring, and it is used in many fields, such as science, engineering, finance, and statistics. In this scenario, we have an urn containing 8 red balls, 3 green balls, and 1 white ball. We are asked to determine the number of ways to draw balls from the urn in various scenarios.

1. To draw one ball of each color, we need to draw one red, one green, and one white ball. We can do this in the following ways:

Choose 1 red ball out of 8, 1 green ball out of 3, and 1 white ball out of 1. This can be done in (8 x 3 x 1) = 24 ways.

Therefore, there are 24 ways to draw one ball of each color.

2. To calculate the number of ways to draw at least one green ball, we need to consider the following scenarios:

Drawing exactly one green ball: We can choose 1 green ball out of 3, and 2 non-green balls out of the remaining 9 (8 red and 1 white). This can be done in (3 x 9C2) = 108 ways.

Drawing exactly two green balls: We can choose 2 green balls out of 3, and 1 non-green ball out of the remaining 9. This can be done in (3C2 x 9) = 27 ways.

Drawing all three green balls: We can choose all 3 green balls out of 3, and no non-green balls. This can be done in 1 way.

Therefore, the total number of ways to draw at least one green ball is the sum of the above three scenarios, which is 108 + 27 + 1 = 136 ways.

3. To draw the same color ball, we need to draw either three red balls, three green balls, or three white balls. We can do this in the following ways:

Choosing 3 red balls out of 8. This can be done in 8C3 = 56 ways.

Choosing 3 green balls out of 3. This can be done in 1 way.

Choosing 3 white balls out of 1. This can be done in 1 way.

Therefore, there are 56 + 1 + 1 = 58 ways to draw the same color ball.

In conclusion, the concept of probability helps us to determine the number of ways that certain events can occur. By using simple combinatorial techniques, we can calculate the probabilities of drawing specific combinations of balls from an urn.

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the constraint ensures that Fund A is limited to a certain proportion of the investment in other funds, indicating that the amount invested in Fund A should be at most 40% of the amount invested in other Funds.

The best interpretation of the constraint A ≤ 0.4(B+xc) is "Amount invested in Fund A should be at most 40% of the amount invested in other Funds."

In this constraint, A represents the amount invested in Fund A, B represents the amount invested in Fund B, and xc represents the total amount invested in Fund C. The expression B+xc represents the total amount invested in Funds B and C combined.

The inequality A ≤ 0.4(B+xc) states that the amount invested in Fund A should be less than or equal to 40% of the total amount invested in Funds B and C. This means that Fund A should not account for more than 40% of the total investment in the other funds.

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The slope of the line perpendicular to the given line is equal to -4/1.

The equation is in the form y=mx+b

so, y=(1/4)x -10

First, find the slope of the line which is the value being multiplied by x. Therefore m=1/4

To find the slope perpendicular to the line you must find the negative reciprocal of 1/4. (Which just means change the sign then flip the fraction)

Multiplying 1/4 by -1 which is -1/4

Then flip the fraction and keep the sign with the numerator to find the reciprocal. So now it’s -4/1

Thus, the slope of the line perpendicular to the given line is equal to -4/1.

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The equation is in the form y=mx+b

so, y=(1/4)x -10

First, find the slope of the line which is the value being multiplied by x. Therefore m=1/4

Multiplying 1/4 by -1 which is -1/4

Then flip the fraction and keep the sign with the numerator to find the reciprocal. So now it’s -4/1

Answer:

x = sqrt (2), y = sqrt (2)

Step-by-step explanation:

Here is how we can approach this problem in a step by step solution:

*Also, if you knew that a 45-45-90 triangle's sides form a ratio of a, a, and sqrt (2) a, you could also use that and substitute in the values to solve. Both ways work! Hope this helps!!

The correct answer is Part C: The ratios calculated in Part B are equivalent, indicating a constant relationship between the number of pairs of jeans and the cost at the local store.Part E: The ratios calculated in Part D are equivalent, indicating a constant relationship between the number of pairs of jeans and the cost at the online store.

Part A:

Table for the local store:

Number of Pairs Cost of Jeans

1 $24

2 $48

3 $72

4 $96

5 $120

Table for the online store:

Number of Pairs Cost of Jeans

1 $22

2 $44

3 $66

4 $88

5 $110

Part B:

To find the ratio of the number of pairs of jeans to the cost at the local store, we divide the number of pairs by the cost:

Number of Pairs Cost of Jeans Ratio

1 $24 1/24 ≈ 0.0417

2 $48 2/48 = 1/24 ≈ 0.0417

3 $72 3/72 = 1/24 ≈ 0.0417

4 $96 4/96 = 1/24 ≈ 0.0417

5 $120 5/120 = 1/24 ≈ 0.0417

The ratio of the number of pairs of jeans to the cost is approximately 0.0417 for each row in the table.

Part C:

The ratios calculated in Part B are equivalent. This means that regardless of the number of pairs of jeans Tim buys at the local store, the ratio of the number of pairs to the cost remains the same. In other words, the relationship between the number of pairs of jeans and the cost at the local store is constant.

Part D:

To find the ratio of the number of pairs of jeans to the cost at the online store, we divide the number of pairs by the cost:

Number of Pairs Cost of Jeans Ratio

1 $22 1/22 ≈ 0.0455

2 $44 2/44 = 1/22 ≈ 0.0455

3 $66 3/66 ≈ 1/22 ≈ 0.0455

4 $88 4/88 = 1/22 ≈ 0.0455

5 $110 5/110 = 1/22 ≈ 0.0455

The ratio of the number of pairs of jeans to the cost is approximately 0.0455 for each row in the table.

Part E:

The ratios calculated in Part D are equivalent. This means that regardless of the number of pairs of jeans Tim buys at the online store, the ratio of the number of pairs to the cost remains the same. In other words, the relationship between the number of pairs of jeans and the cost at the online store is constant.

Part F:

To calculate the cost of the jeans after sales tax is applied, we need to add 7% of the cost to the original cost:

Number of Pairs Cost of Jeans Sales Tax (7%) Cost with Sales Tax

1 $24 $1.68 $25.68

2 $48 $3.36 $51.36

3 $72 $5.04 $77.04

4 $96 $6.72 $102.72

5 $120 $8.40 $128.40

Part G:

To find the ratio of the number of pairs of jeans to the cost with sales tax, we divide the number of pairs by the cost with sales tax:

Number of Pairs Cost with Sales Tax Ratio

1 $25.68 1/25.68 ≈ 0.0389

2 $51.36 2/51.36 ≈ 0.0389

3 $77.04 3/77.04 ≈ 0.0389

4 $102.72 4/102.72 ≈ 0.0389

5 $128.40 5/128.40 ≈ 0.0389

The ratio of the number of pairs of jeans to the cost with sales tax is approximately 0.0389 for each row in the table.

Part H:

The ratios calculated in Part G are equivalent. This means that regardless of the number of pairs of jeans Tim buys at the local store with sales tax applied, the ratio of the number of pairs to the cost remains the same. In other words, the relationship between the number of pairs of jeans and the cost with sales tax applied at the local store is constant.

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The Hessian matrix for the second function is;

∇²f(x₁, x₂) = [ -3x₂/(4x₁³x₂²), 3/(4√(2x₁³x₂³)); 3/(4√(2x₁³x₂³)), -3x₁/(4x₁²x₂³) ]

Gradient vector: Gradient vector is defined as the vector of the partial derivatives of a function with respect to its independent variables.

The gradient vector is represented by the symbol ∇. It is calculated as follows;

For (1) given above; f(x1, x2) = (2x1 + 3x2)ln(x2+x1)

Thus;∂f/∂x₁ = (2 + 3x₂/(x₁ + x₂))ln(x₁ + x₂)∂f/∂x₂

= (3x₁ + 2x₂)/(x₁ + x₂) ln(x₁ + x₂)

Thus the gradient vector for the first function is;

∇f(x₁, x₂) = [ (2 + 3x₂/(x₁ + x₂))ln(x₁ + x₂), (3x₁ + 2x₂)/(x₁ + x₂) ln(x₁ + x₂) ]

For the second function, (2) given above;

f(x₁, x₂) = 3√(2x₁x₂)

Thus;∂f/∂x₁ = 3/2 x₂/√(2x₁x₂)∂f/∂x₂ = 3/2 x₁/√(2x₁x₂)

Thus the gradient vector for the second function is;

∇f(x₁, x₂) = [ 3/2 x₂/√(2x₁x₂), 3/2 x₁/√(2x₁x₂) ]

Hessian Matrix: Hessian matrix is defined as a square matrix of second-order partial derivatives of a scalar-valued function. It is used to determine the type of critical point.

To calculate the Hessian matrix of a function, you differentiate each element of the gradient vector by all the independent variables of the function. It is represented by the symbol ∇².

For the first function, given in (1) above, the gradient vector is;

∇f(x₁, x₂) = [ (2 + 3x₂/(x₁ + x₂))ln(x₁ + x₂), (3x₁ + 2x₂)/(x₁ + x₂) ln(x₁ + x₂) ]

Differentiating each element of the gradient vector by all the independent variables of the function, we get;

∂²f/∂x₁² = -3x₂/(x₁ + x₂)² ln(x₁ + x₂) + (2 + 3x₂/(x₁ + x₂))(1/(x₁ + x₂))∂²f/∂x₂²

= (3x₁ + 2x₂)/(x₁ + x₂)² ln(x₁ + x₂) - ((3x₁ + 2x₂)/(x₁ + x₂))²∂²f/∂x₁∂x₂

= ∂²f/∂x₂∂x₁

= 3/(x₁ + x₂) ln(x₁ + x₂) + 3x₂/(x₁ + x₂)²∂²f/∂x₁∂x₂

= ∂²f/∂x₂∂x₁

Thus the Hessian matrix for the first function is;

∇²f(x₁, x₂) = [ -3x₂/(x₁ + x₂)² ln(x₁ + x₂) + (2 + 3x₂/(x₁ + x₂))(1/(x₁ + x₂)), 3/(x₁ + x₂) ln(x₁ + x₂) + 3x₂/(x₁ + x₂)²; 3/(x₁ + x₂) ln(x₁ + x₂) + 3x₂/(x₁ + x₂)², (3x₁ + 2x₂)/(x₁ + x₂)² ln(x₁ + x₂) - ((3x₁ + 2x₂)/(x₁ + x₂))² ]

For the second function given in (2) above, the gradient vector is; ∇f(x₁, x₂) = [ 3/2 x₂/√(2x₁x₂), 3/2 x₁/√(2x₁x₂) ]Differentiating each element of the gradient vector by all the independent variables of the function, we get;

∂²f/∂x₁² = -3x₂/(4x₁³x₂²)∂²f/∂x₂²

= -3x₁/(4x₁²x₂³)∂²f/∂x₁∂x₂

= ∂²f/∂x₂∂x₁

= 3/(4√(2x₁³x₂³))

Thus the Hessian matrix for the second function is;

∇²f(x₁, x₂) = [ -3x₂/(4x₁³x₂²), 3/(4√(2x₁³x₂³)); 3/(4√(2x₁³x₂³)), -3x₁/(4x₁²x₂³) ]

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To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. To find the conjugate, just change the sign in the denominator.

Now, According to the question:

The complex number is basically the combination of a real number and an imaginary number. The complex number is in the form of a+ib, where a = real number and ib = imaginary number. Also, a,b belongs to real numbers and i = √-1.

Hence, a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. To find the conjugate, just change the sign in the denominator.

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