Consider the following function.

g(x, y) = e^(− 4x^2 + 7y^2 + 14√8y

(a) Find the critical point of g.
If the critical point is (a, b) then enter 'a,b' (without the quotes) into the answer box.
(b) Using your critical point in (a), find the value of D(a, b) from the Second Partials test that is used to classify the critical point.
(c) Use the Second Partials test to classify the critical point from (a).

The antiderivative of the given function is: [tex]-3/x^2 + 8ln|x| + C[/tex], where x is not equal to 0.

What is antiderivative?

Antiderivative is the reverse process of differentiation in calculus. Given a function f(x), an antiderivative of f(x) is another function F(x) whose derivative is equal to f(x).

The given function can be written as:

[tex](9x^3+8x^5)/x^6 = 9/x^3 + 8/x[/tex]

To find the antiderivative, we integrate each term separately using the power rule of integration:

∫(9/[tex]x^3[/tex]) dx = -3/[tex]x^2[/tex] + C1

and

∫(8/x) dx = 8ln|x| + C2

where C1 and C2 are constants of integration.

Therefore, the antiderivative of the given function is:

∫([tex]9x^3+8x^5[/tex])/[tex]x^6 dx[/tex] = ∫([tex]9/x^3[/tex]) dx + ∫(8/x) dx

= ([tex]-3/x^2 + C1[/tex]) + (8ln|x| + C2)

= [tex]-3/x^2[/tex] + 8ln|x| + C

where C = C1 + C2 is a constant of integration. Therefore, the antiderivative of the given function is:

[tex]-3/x^2 + 8ln|x| + C[/tex], where x is not equal to 0.

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Answer:

r≈3.39

Step-by-step explanation:

I'm not quite sure but it's the best I can do.

Answer:

radius of base = 6 cm

Step-by-step explanation:

In order to calculate the radius of the base of a cone given its volume and height, we have to use the formula for the volume of a cone:

[tex]\boxed{\mathrm{V = \frac{1}{3} \pi r^2h}}[/tex],

where:

• V ⇒ volume of the cone

• r ⇒ radius of the base of the cone

• h ⇒ height of the cone

The question gives us the value of the volume of the cone (72π cm³) as well as its height (6 cm). By substituting these values into the equation above, we can solve for r to get the radius of the base of the cone:

[tex]\mathrm{V = \frac{1}{3} \pi r^2h}[/tex]

⇒ [tex]72 \pi = \frac{1}{3} \times \pi \times r^2 \times 6[/tex]

⇒ [tex]r^2 = \frac{72 \pi}{2 \pi}[/tex]

⇒ [tex]r = \sqrt{\frac{72 \pi}{2 \pi}}[/tex]

⇒ [tex]r = \sqrt{36}[/tex]

⇒ r = 6 cm

Therefore, the radius of the base of the cone is 6 cm.

The original price of the bracelet was £1400.

Percentage is a way of expressing a proportion or a fraction as a part of 100. It is denoted using the symbol "%". For example, 50% means 50 out of 100, or half, while 25% means 25 out of 100, or one-quarter.

We can start by using the information given to set up an equation that relates the original price of the bracelet with the sale price and the percentage reduction:

original price x (100% - 70%) = sale price

Simplifying the percentage reduction:

original price x 30% = sale price

Substituting the given sale price (£420):

original price x 30% = £420

To solve for the original price, we need to isolate it on one side of the equation. We can do this by dividing both sides by 30% (which is the same as multiplying by 100/30 or 10/3):

original price = sale price / 30% = £420 / 30% = £1400

Therefore, the original price of the bracelet was £1400.

Complete question - A jewelry shop is having a sale. A bracelet is now reduced to £420. This is 70% of the original price. Work out the original price of the bracelet.

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Answer:

150 cubic meters

Step-by-step explanation:

A rectangular prism is a box. The volume of a rectangular prism is:

Volume=

length×width×height

All three of these measures are given, so we just fill them in.

Vol = 10 × 3 × 5

= 150 cubic meters

The final equation of the exponential function is [tex]y = 27/3^x[/tex], which passes through the points (2,3) and (5,1/9).

When mοdelling a cοnnectiοn, expοnential functiοns simulate hοw a cοnstant change in the independent variable results in the same prοpοrtiοnal change. The expοnentiaI functiοn is a mathematicaI functiοn denοted by f(x)=[tex]\exp[/tex] or [tex]e^{x}.[/tex]

Let's start by plugging in the values οf the first pοint (2,3) intο the equatiοn:

3 = ab²

Similarly, we can plug in the values οf the secοnd pοint (5,1/9):

1/9 = ab^5

Sοlving fοr a in the first equatiοn:

a = 3/b²

Substituting a in the secοnd equatiοn:

1/9 = (3/b²) * b^5

Simplifying:

1/9 = 3b³

b^3 = 1/27

b = 1/3

Substituting b intο the first equatiοn tο sοlve fοr a:

3 = a(1/3)²

3 = a/9

a = 27

Therefοre, the equatiοn οf the expοnential functiοn is y = 27(1/3)^x οr y = 27/3^x.

So the final equation of the exponential function is [tex]y = 27/3^x[/tex], which passes through the points (2,3) and (5,1/9).

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The inverse function:  [tex]f^{-1} (x) =[/tex] [tex](\frac{x -5}{5} )^{2} -13[/tex]

The inverse is defined on the domain x < 5 and x ≥ -13 for the original function, which means that the range of the original function is y ≥ 5.

A function is a relationship that exists between two sets of numbers, with each input from the first set, known as the domain, corresponding to only one output from the second set, known as the range.

Given function is;   [tex]f(x) = 5\sqrt{(x + 13)} + 5[/tex]

To find the inverse of the given function, we first replace f(x) with y:

⇒  [tex]y = 5\sqrt{(x + 13)} + 5[/tex]

Subtract 5 from both sides:

⇒ [tex]y -5 = 5\sqrt{(x + 13)}[/tex]

⇒ [tex]\frac{(y -5)}{5} = \sqrt{(x + 13)}[/tex]

⇒ [tex](\frac{y -5}{5} )^{2} = x + 13[/tex]

⇒ [tex](\frac{y -5}{5} )^{2} -13 = x[/tex]

Now we have x in terms of y, so we can replace x with f⁻¹(x) and y with x to get the inverse function:

f⁻¹(x) = [tex](\frac{x -5}{5} )^{2} -13[/tex]

The domain of the inverse function is x ≥ 5, because this is the range of the original function, and we were given that the inverse is defined on the domain x < 5. However, we must also exclude the value x = 5, because the denominator of the fraction [tex](\frac{x -5}{5} )^{2}[/tex] becomes zero at this value. Therefore, the domain of f⁻¹(x) is x > 5.

We were given that x ≥ -13 for the original function, which means that the range of the original function is y ≥ 5. Therefore, the domain of the inverse function becomes the range of the original function, and the range of the inverse function becomes the domain of the original function.

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The ninth term of the geometric sequence with a1=8 and [tex]r = \frac{1}{2}[/tex] is [tex]\frac{3}{2}[/tex]. The term equal to 36 is the fourth term, the term equal to 32 is the third term, and the common ratio is 1/4.

What are the common ratio of the geometric sequence with a1=8 and r=1/2?

The general formula for the nth term of a geometric sequence is given by:

[tex]a_n &= a_1 \times r^{n-1}[/tex]

a) To find the ninth term of the sequence with a1=8, r=1/2, and n=9, we can plug in these values into the formula:

[tex]a_9 &= a_1 \times r^{9-1} \\&= 8 \times \left(\frac{1}{2}\right)^{9-1} \\&= 8 \times \left(\frac{1}{2}\right)^8 \\&= 8 \times \frac{1}{256} \\&= \frac{1}{32}[/tex]

So the ninth term is 1/32.

b) To find the term that is equal to 36, we can set the formula equal to 36 and solve for n:

[tex]a_n &= a_1 \times r^{n-1} = 36 \\a_1 \times \left(\frac{1}{2}\right)^{n-1} &= 36 \\8 \times \left(\frac{1}{2}\right)^{n-1} &= 36 \\\left(\frac{1}{2}\right)^{n-1} &= \frac{36}{8} \\\left(\frac{1}{2}\right)^{n-1} &= 4.5 \\n-1 &= \log_{1/2}(4.5) \\[/tex]

n-1 = 3.17 (rounded to two decimal places)

n = 4.17 (rounded to two decimal places)

Therefore, the term that is equal to 36 is the fourth term.

c) To find the term that is equal to 32, we can set the formula equal to 32 and solve for n:

[tex]a_n &= a_1 \times r^{n-1} = 32 \\a_1 \times \left(\frac{1}{2}\right)^{n-1} &= 32 \\8 \times \left(\frac{1}{2}\right)^{n-1} &= 32 \\\left(\frac{1}{2}\right)^{n-1} &= 4 \\n-1 &= \log_{1/2}(4) \\[/tex]

n-1 = 2

n = 3

Therefore, the term that is equal to 32 is the third term.

d) To find the common ratio, we can use the formula:

[tex]r &= \frac{a_n}{a_{n-1}}[/tex]

where an is the nth term and a(n-1) is the term before it. For this problem, we are given a1=8 and n=9, so we can use the formula to find a9 and a8, and then calculate the ratio:

[tex]a_9 &= a_1 \times r^{n-1} = 8 \times (1/2)^8 = \frac{1}{256} \\[/tex]

[tex]a_8 &= a_1 \times r^{n-2} = 8 \times (1/2)^7 = \frac{1}{64} \\[/tex]

[tex]r &= \frac{a_9}{a_8} = \frac{\frac{1}{256}}{\frac{1}{64}} = \frac{1}{4}[/tex]

Therefore, the common ratio is 1/4.

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Answer:

9.6 / 0.91 ≈ 10.55

Step-by-step explanation:

I used a calculator

Answer:

X = 115

Step-by-step explanation:

Thus, the cardinal number of the given set A is found as: n(A) = 4.

Cardinal numbers are those that are used to count things or actual items. They are also referred to as "cardinals" or "counting numbers."

Given set:

set A = {2, 5, 6 , 8}.

n(A)  shows the cardinal number for the set A, which is the total number of elements in the set.

n(A) = 4

Thus, the cardinal number of the given set A is found as: n(A) = 4.

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Complete question:

Find cardinal number n(A) for the set A = {2, 5, 6 , 8}.

Using operations, we can find the values:

Option D. Customer will pay = $41.26

Option A. Customer pays for T-shirt = $34.82

A set of guidelines that must be followed in a specific order in order to solve an equation is known as the order of operations. The term "operations" in mathematics is used to refer to the process of evaluating any mathematical expression, which includes arithmetic operations like addition, subtraction, multiplication, and division.

If it costs the shop $16.88, we will add $3.75 and double it.

2(16.88 + 3.75) = $41.26

Next part,

If it costs the shop $12.99, we will add $3.75 and double it.

2(12.99 + 3.75) = $33.48

Next, we will add the sales tax to this amount.

$33.48 + $1.34 = $34.82

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Answer:

[tex]x = y = \frac{5 \sqrt{2} }{2} [/tex]

Step-by-step explanation:

Use trigonometry:

[tex] \sin(45°) = \frac{x}{5} [/tex]

Cross-multiply to find x:

[tex]x = 5 \times \sin(45°) = 5 \times \frac{ \sqrt{2} }{2} = \frac{5 \sqrt{2} }{2} [/tex]

Use the Pythagorean theorem to find y:

[tex] {y}^{2} = {5}^{2} - {x}^{2} [/tex]

[tex] {y}^{2} = {5}^{2} - ( { \frac{5 \sqrt{2} }{2}) }^{2} = 25 - \frac{25 \times 2}{4} = \frac{25}{1} - \frac{50}{4} = \frac{25 \times 4}{4} - \frac{50}{4} = \frac{100}{4} - \frac{50}{4} = \frac{50}{4} = \frac{25}{2} [/tex]

[tex]y > 0[/tex]

[tex]y = \sqrt{ \frac{25}{2} } = \frac{5 \sqrt{2} }{2} [/tex]

A value 3 standard deviations above the mean is 110

a. To find a value that is 3 standard deviations above the mean, we can use the formula:

value = mean + (number of standard deviations) x standard deviation

So, substituting the given values, we get:

value = 77 + (3) x 11

value = 77 + 33

value = 110

Therefore, a value 3 standard deviations above the mean is 110.

b. To find a value that is 2.5 standard deviations below the mean, we can use the same formula:

value = mean - (number of standard deviations) x standard deviation

Substituting the given values, we get:

value = 77 - (2.5) x 11

value = 77 - 27.5

value = 49.5

Therefore, a value 2.5 standard deviations below the mean is 49.5.

c. To find a value that is 2 standard deviations below the mean, we can use the same formula:

value = mean - (number of standard deviations) x standard deviation

Substituting the given values, we get:

value = 77 - (2) x 11

value = 77 - 22

value = 55

Therefore, a value 2 standard deviations below the mean is 55.

d. To find a value that is 1 standard deviation above the mean, we can use the same formula:

value = mean + (number of standard deviations) x standard deviation

Substituting the given values, we get:

value = 77 + (1) x 11

value = 77 + 11

value = 88

Therefore, a value 1 standard deviation above the mean is 88.

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The exact values for trigonometric functions are:

a) cos(A+B)=56/65

b) sin(A+B)=-33/65

c) sin(A-B)=-33/65

a) We have, cos A = -3/5 (using Pythagorean identity

and sin B = -5/13 (using Pythagorean identity  = 1).

Using the formula for cosine of sum of two angles, we have:

cos (A+B) = cos A cos B - sin A sin B

[tex]=(-3/5)(-12/13)-(4/5)(-5/13)\\\\=56/65[/tex]

b) Using the same formula for sine of sum of two angles, we have:

sin (A+B) = sin A cos B + cos A sin B

[tex]=(-3/5)(-12/13)-(4/5)(-5/13)\\\\=-33/65[/tex]

c) Using the formula for sine of difference of two angles, we have:

sin (A-B) = sin A cos B - cos A sin B

[tex]453=(4/5)(-12/13)+(-3/5)(-5/13)\\\\=-33/65[/tex]

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Answer:

Got you bro

Step-by-step explanation:

The two-dimensional shape appears to be a semi-circle, and it is being rotated about a line to form a three-dimensional shape. The resulting shape is a sphere.

Answer:

Step-by-step explanation:

The two-dimensional shape appears to be a semi-circle, and it is being rotated about a line to form a three-dimensional shape.

The resulting shape is a sphere.

The correct answer of the following is option A which is 49+32+m =180 because an angle on a straight line is equal to 180 degree.

what is angle?

Angles are an important concept in mathematics, physics, and engineering. An angle is formed by two rays or lines that share a common endpoint called a vertex. The measurement of an angle is usually expressed in degrees or radians.

In geometry, angles are classified according to their measure. An acute angle measures less than 90 degrees, a right angle measures exactly 90 degrees, an obtuse angle measures greater than 90 degrees but less than 180 degrees, and a straight angle measures exactly 180 degrees.

Angles can be formed by lines that intersect or by shapes such as triangles, quadrilaterals, and circles. The study of angles is an important part of trigonometry, which is the branch of mathematics concerned with the relationships between angles and the sides and heights of triangles.

Angles are also used in physics to describe the orientation and motion of objects. For example, the angle between two vectors can be used to calculate the direction of a force or the trajectory of a moving object.

In engineering, angles are used to design structures such as bridges, buildings, and machines. Engineers use trigonometry to calculate the angles and lengths of the various components of these structures to ensure they are safe and structurally sound.

Overall, angles are a fundamental concept in mathematics and have numerous applications in science, technology, and everyday life.

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The width of the coal tray is approximately 16.24 inches.

The width of the coal tray is determined as follows:

The given formula is P α 1/(1 + d²).

We can rewrite it as P = k/(1 + d²), where k is a constant of proportionality.

Since the problem states that the width of the coal tray is equal to d, we can assume that the width of the tray is equal to 1 (arbitrary units), without loss of generality.

So, we have P = k/(1 + d²) = k/(1 + (distance from food to coals / 1)²)

P = k/(1 + distance from food to coals²)

When the food is 16 inches above the tray, the distance from food to coals is d = 16/1 = 16.

When P = 0.53, we have:

0.53 = k/(1 + 16²)

k = 0.53(1 + 16²)

k ≈ 139.88

Now we can use the equation P = 0.53 = 139.88/(1 + d²) to solve for d:

0.53(1 + d²) = 139.88

1 + d² = 264.45

d² = 263.45

d ≈ 16.24

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Answer:

A = (1/2)(20 yd)(7.75 yd) = 77.5 square yards

Answer:

x = 3√3;

y = 3

Step-by-step explanation:

Use trigonometry:

[tex] \sin(30°) = \frac{y}{6} [/tex]

Cross-multiply to find y:

[tex]y = 6 \times \sin(30°) = 6 \times 0.5 = 3[/tex]

Use the Pythagorean theorem to find x:

[tex] {x}^{2} = {6}^{2} - {y}^{2} [/tex]

[tex] {x}^{2} = {6}^{2} - {3}^{2} = 36 - 9 = 27[/tex]

[tex]x > 0[/tex]

[tex]x = \sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3} [/tex]

The inverse of the function f(x) is f⁻¹(x) = √(x) - 8

To find the inverse of the function f(x) = (x + 8)², we need to solve for x in terms of y:

y = (x + 8)²

Taking the square root of both sides, we get:

±√(y) = x + 8

Solving for x, we get:

x = ±√(y) - 8

Since we want the inverse function to be a function (i.e., have a unique output for each input), we must choose the positive square root. Therefore, the inverse function is:

f⁻¹(x) = √(x) - 8

The domain of f⁻¹ is the range of f, which is [0, ∞). Therefore, the domain of f⁻¹ is [0, ∞).

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Using simple interest, we can find the value of interest here to be $162 per month.

Calculating the amount of interest that will be owed on a sum of money at a certain rate and for a specific period of time is possible using simple interest. The principal amount in the case of simple interest does not change, in contrast to compound-interest, where the interest is added to the principal to calculate the principal for the new principal for the following year.

Given in the question,

Principle, P = $18000

Rate of interest, R = 0.45%

= 0.45/100

Time in years, T = 24 months

= 2 years.

I = P R T

= 18000 × 0.45/100 × 2

= $162

Therefore, interest here is $162 per month.

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The tangent of 60° is √3, so the coterminal angle of tan(780°) is √3.

Coterminal angles are angles that have the same initial and terminal sides in standard position.

To find the coterminal angle of tan(780°), we need to add or subtract multiples of 360° to 780° until we get an angle between 0° and 360°, because angles that differ by a multiple of 360° have the same trigonometric function values.

First, we can subtract 360° from 780°:

780° - 360° = 420°

This is not yet between 0° and 360°, so we can subtract another 360°:

420° - 360° = 60°

Now we have an angle between 0° and 360° that is coterminal with 780°.

The tangent function has a period of 180°, which means that the tangent function has the same value for angles that differ by a multiple of 180°. Since 60° is an acute angle, we can use the tangent of 60° to find the tangent of 780°:

tan(780°) is equivalent to tan(780° - 720°) = tan(60°)

The tangent of 60° is √3, so the coterminal angle of tan(780°) is √3.

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The required statement is "The sky is clear if and only if the plane is on time."

The statement "Q<->P" is a logical statement that uses the biconditional operator "<->" which means "if and only if." This operator connects two propositions in such a way that both propositions are true or false together.

In this case, the propositions are "Q" and "P" which are defined as "The sky is clear" and "The plane is on time," respectively. Therefore, the statement "Q<->P" can be translated into words as "The sky is clear if and only if the plane is on time."

This means that the truth of the proposition "Q" (the sky is clear) is dependent on the truth of proposition "P" (the plane is on time) and vice versa.

If the plane is on time, then the sky must be clear, and if the sky is clear, then the plane must be on time. If either of these propositions is false, then the bi-conditional statement is false as well.

Thus, required statement is "The sky is clear if and only if the plane is on time."

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The population will exceed 2458 after approximately 10.1465 units of time, where the time unit depends on the context of the problem (e.g., hours, days, etc.).

A statement proving the equality of two expressions is known as an equation. It can include variables, integers, and mathematical operations like addition, subtraction, multiplication, and division. It also incorporates mathematical symbols. In mathematics and science, equations are frequently used to illustrate connections between quantities. The equals sign (=) is typically used in equations to denote that the expressions on each side of the sign have the same value. For instance, the formula 2 + 3 = 5 demonstrates that the total of 2 and 3 equals 5. Equations can be solved to determine a variable's value or to determine if a certain value meets the connection that the equation describes.

To estimate when the population will exceed 2458, we can set up an inequality using the equation for the population growth:

p(t) > 2458

Substituting the given equation for p(t), we get:

[tex]1100e^0.12t[/tex] > 2458

Dividing both sides by 1100, we get:

[tex]e^0.12t > 2.23545[/tex]

Taking the natural logarithm of both sides, we get:

0.12t > ln(2.23545)

Solving for t, we get:

t > ln(2.23545)/0.12

Using a graphing calculator to evaluate this expression, we get:

t > 10.1465

Therefore, the population will exceed 2458 after approximately 10.1465 units of time, where the time unit depends on the context of the problem (e.g., hours, days, etc.).

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The product of 3 and 10v√40 in simplest radical form is 30v(2√5).

What is product?

To find the product of 3 and 10v√40 in simplest radical form, we can simplify the radical first.

First, we can simplify 40 by finding its prime factorization:

40 = 2 × 2 × 2 × 5

Next, we can rewrite 10v√40 as 10v√(2 × 2 × 2 × 5) to separate out the perfect squares:

10v√(2 × 2 × 2 × 5) = 10v(√2 × √2 × √2 × √5)

We can then simplify the perfect squares under the radical:

10v(√2 × √2 × √2 × √5) = 10v(2√5)

Now we can multiply 3 and 10v(2√5):

3 × 10v(2√5) = 30v(2√5)

So the product of 3 and 10v√40 in simplest radical form is 30v(2√5).

What is prime factorization?

Prime factorization is the process of expressing a composite number as a product of its prime factors. In other words, it is finding the prime numbers that can be multiplied together to get the original number. For example, the prime factorization of 24 is 2 x 2 x 2 x 3 or 2³ x 3, since 24 can be expressed as a product of the prime numbers 2 and 3, and each of these primes is repeated as many times as necessary to get the original number. Prime factorization is an important concept in mathematics and has many practical applications, including in cryptography, number theory, and computer science.

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Answer:

[tex]x = 4 \frac{2}{3} [/tex]

[tex]y = 5[/tex]

Step-by-step explanation:

Multiply the second equation by -5 to eliminate 15x:

{15x - 4y = -50,

{3x - 2y = -16; / × (-5)

+ {15x - 4y = -50,

{-15x + 10y = 80;

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

6y = 30 / : 6

y = 5

Make x the subject from the 2nd equation (it doesn't matter, you can do it from the 1st one instead):

15x = -50 + 4y / : 15

[tex]x = 3 \frac{1}{3} + \frac{4}{15} y[/tex]

[tex]x = 3 \frac{1}{3} + \frac{4}{15} \times 5 = \frac{14}{3} = 4 \frac{2}{3} [/tex]

Therefore, the absolute maximum value of f(x) over (-2, 3) is 201 and it occurs at x = -2. The absolute minimum value of f(x) over (-2, 3)is -98 and it occurs at x = 3.

We must identify the crucial points of the function within the period in order to determine the function's absolute maximum and minimum values.

Define critical point?

The critical points are those where the function's derivative is zero or undefinable. The function is then assessed at these pivotal points as well as the interval's endpoints.

Absolute maximum and lowest values are represented by the largest and smallest values, respectively.

The derivative of f(x) = 3x⁴ - 4x³ - 12x² + 1 over (-2, 3) is initially found as follows:

f'(x) = 12x³ - 12x²- 24x

If we set f'(x) to 0, we obtain:

12x³ - 12x² - 24x = 0

By multiplying both sides of this equation by 12x, we may simplify it to:

x²- x - 2 = 0

The answer to this quadratic equation is:

x = -1, x = 0, x = 2

Now we evaluate f(x) at these critical points and at the endpoints of the interval:

f(-2) = 201

f(-1) = -6

f(0) = 1

f(2) = 49

f(3) = -98

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Answer: For Damian's investment:

The interest rate is 3%, compounded quarterly, which means the interest rate per quarter is 3%/4 = 0.75%.

The number of quarters in 16 years is 16*4 = 64.

Using the formula for compound interest, the balance after 16 years is:

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

where:

P = the principal (initial investment) = $81,000

r = the interest rate per quarter = 0.75%

n = the number of times the interest is compounded per year = 4 (quarterly)

t = the number of years = 16

A = 81000*(1 + 0.0075/4)^(4*16) = $157,222.39

For Marques's investment:

The interest rate is 2%, compounded continuously.

Using the formula for continuous compound interest, the balance after 16 years is:

A = Pe^(rt)

where:

P = the principal (initial investment) = $81,000

r = the interest rate per year = 2%

t = the number of years = 16

A = 81000e^(0.0216) = $131,518.16

Therefore, Damian would have $157,222.39 - $131,518.16 = $25,704.23 more than Marques in his account after 16 years. Rounded to the nearest dollar, this is $25,704.

Step-by-step explanation:

Damian would have approximately $351 more in his account than Marques after 16 years.

Interest rate is the percentage of a loan or deposit that is charged as interest or earned as interest over a period of time. It is expressed as a percentage of the principal amount borrowed or deposited, and it represents the cost of borrowing or the reward for saving money.

We can use the compound interest formula to calculate the future value of each investment after 16 years and then subtract to find the difference.

For Damian's investment, the interest rate is 3% per year, compounded quarterly. This means that the quarterly interest rate is r = 0.03/4 = 0.0075, and the number of compounding periods is n = 16 x 4 = 64. The future value of Damian's investment is:

F = 81000 * [tex](1 + r)^n[/tex]

  = 81000 * [tex](1.0075)^64[/tex]  

  = 129,535.28

For Marques's investment, the interest rate is 2% per year, compounded continuously. This means that the continuously compounded interest rate is r = 0.02, and the number of compounding periods is n = 16 x 1 = 16. The future value of Marques's investment is:

F = 81000 * [tex]e^(rn)[/tex]

  = 81000 * [tex]e^(0.0216)[/tex]

  = 129,183.81

The difference between the two investments is:

129,535.28 - 129,183.81 = 351.47

So Damian would have approximately $351 more in his account than Marques after 16 years. Rounded to the nearest dollar, the difference is $351.

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Rolle’s Theorem can be applied to the closed interval and the value of x = (12 ±√12)/3

Rolle's theorem states that "If a function f is defined in the closed interval [a, b] in such a way that it satisfies the following condition: If is continuous οn [a, b], ii) f is differentiable οn (a, b), and iii) f (a) = f (b), then there exists at leastοne value οf x, let us assume this value to be c, which lies between a and b i.e. (a < c < b) in such a way that f'(c) = 0.".

Here, we have

Given: f(x) = (x-1)(x-2)(x-3), [1,3]

We have to determine whether Rolle’s Theorem can be applied to the closed interval.

This function is continuous in [1, 3] and is differentiable everywhere except at the points x = 1, 2, 3.

This point is in the interval [1, 3], and since Rolle's Theorem requires that the function must be differentiable on the open interval (1, 3).

f(x) = (x-1)(x-2)(x-3)

f'(x) = (x-2)(x-3) + (x-1)(x-3) + (x-1)(x-2)

f'(x) = x² - 5x + 6 + x² - 4x + 3 + x² -3x + 2

f'(x) = 3x² -12x + 11

f'(x) = 0

3x² -12x + 11 = 0

x = (12 ±√12)/3

Hence, Rolle’s Theorem can be applied to the closed interval.

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