Consider the function f : R→R
given by f(x) = x^(4) - 32 x^(2)
Find the minimum and and maximum values (absolute) of f over the interval [3,5]. Provide the exact values. Minimum value a Maximum value.

The possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

The Rational Root Theorem states that if a polynomial [tex]f(x) = anxn + an-1xn-1 + ... + a1x + a0[/tex] has a rational root, then it must be of the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.

For the given polynomial [tex]f(x) = -x + 14x3 - 18x2 - 4x5 - 26x4 + 8[/tex], the constant term is 8 and the leading coefficient is -4.

The factors of 8 are ±1, ±2, ±4, and ±8. The factors of -4 are ±1, ±2, and ±4.

Therefore, the possible rational roots of f(x) are:

p/q = ±1/1, ±2/1, ±4/1, ±8/1, ±1/2, ±2/2, ±4/2, ±8/2, ±1/4, ±2/4, ±4/4, ±8/4

Simplifying gives us the possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±4/2, ±8/2, ±1/4, ±2/4, ±8/4

Simplifying further gives us the final list of possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±2, ±4, ±1/4, ±1/2, ±2

Removing duplicates, the final list of possible rational roots is:

±1, ±2, ±4, ±8, ±1/2, ±1/4

Therefore, the possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

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Hence, the total surface area of the rectangular prism is 736 in².

A 3D object's surface area is calculated as the total of all of its faces. For instance, if we wanted to figure out how much paint we would need to paint a cube, we would use the cube's surface area.

To calculate the total surface area of the rectangular prism, we need to find the area of each of its six faces and add them together.

The area of the top and bottom faces, which are identical, is:

Length x Breadth = 16 in x 8 in = 128 in².

The area of the front and back faces, which are also identical, is:

Height x Breadth = 10 in x 8 in = 80 in².

The area of the left and right faces, which are also identical, is:

Height x Length = 10 in x 16 in = 160 in².

Therefore, the total surface area of the rectangular prism is:

2(Length x Breadth) + 2(Height x Breadth) + 2(Height x Length)

= 2(16 in x 8 in) + 2(10 in x 8 in) + 2(10 in x 16 in)

= 256 in² + 160 in² + 320 in²

= 736 in²

Hence, the total surface area of the rectangular prism is 736 in².

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Answer:

[tex]x=\frac{3}{2} - i\frac{\sqrt{7}}{2}[/tex]

[tex]x=\frac{3}{2} + i\frac{\sqrt{7}}{2}[/tex]

Step-by-step explanation:

We can use the quadratic formula to solve the equation:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 1, b = -3, and c = 4. Substituting these values into the formula, we get:

x = (3 ± sqrt((-3)^2 - 4(1)(4))) / 2(1)

x = (3 ± sqrt(9 - 16)) / 2

x = (3 ± sqrt(-7)) / 2

Since the square root of a negative number is not a real number, the solutions to the quadratic equation are complex numbers. Therefore, the answer is:

[tex]x=\frac{3}{2}+i\frac{\sqrt{7} }{2}[/tex]

[tex]x=\frac{3}{2} - i\frac{\sqrt{7}}{2}[/tex]

The distance traveled by the bullet is 1000 feet.

The distance traveled by the bullet can be calculated by multiplying its speed by the time it takes to hit the target:

Distance = Speed x Time

Substituting the given values, we get:

[tex]Distance = 510^6 feet/hour * 210^{-4} hours[/tex]

Simplifying this expression, we get:

Distance = 1000 feet

Therefore, the bullet has traveled 1000 feet before striking the target.

In summary, the distance traveled by the bullet can be calculated by multiplying its speed by the time it takes to hit the target. In this case, the distance traveled is 1000 feet.

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The composition of the mixture is: 7.323 pounds of the variety priced at $5.77 per pound and 2.677 pounds of the variety priced at $8.87 per pound.

To solve this problem, we can use a system of equations. Let x be the number of pounds of the first variety of birdseed, and y be the number of pounds of the second variety of birdseed.

The first equation will represent the total weight of the mixture:
x + y = 10

The second equation will represent the total cost of the mixture:
5.77x + 8.87y = 6.70(10)

Now we can use the elimination method to solve for one of the variables. Multiply the first equation by -5.77 to eliminate the x variable:
-5.77x - 5.77y = -57.7

Add this equation to the second equation:
3.1y = 8.3

Solve for y:
y = 2.677

Now we can plug this value back into the first equation to solve for x:
x + 2.677 = 10
x = 7.323

So, there are 7.323 pounds of the first variety of birdseed and 2.677 pounds of the second variety of birdseed in the mixture.

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When he rests for a few hours after coming back from ice-store he rests for 2 hours the time the plane takes off is 9:10 pm.

Total distance traveled is 151 miles.

Dave spends 25 + 10 + 55 + 25 = 115 minutes (or 1 hour and 55 minutes) on stops.

Dave rests for 2 hours.

To calculate the time the plane takes off, we need to add up all the time Dave spends on driving, stops, and rests, and then add that time to the time he starts (12:45 pm).

Time to drive from A to C and back to A: 2 * 6 / 60 = 0.2 hours

Time to drive from A to C to D to B to A: (6 + 24 + 24 + 3) / 60 = 0.95 hours

Time to drive from A to C to D to home to E to market to E: (6 + 24 + 24 + 40 + 5 + 5) / 60 = 2.33 hours

Time to rest: 2 hours

Total time spent: 0.2 + 0.95 + 2.33 + 1.92 = 5.4 hours

Time of departure: 12:45 pm + 5.4 hours = 6:25 pm

Adding 45 minutes for the luggage check and boarding, the plane takes off at 6:25 pm + 0.45 hours = 7:10 pm

Adding another 2 hours for the time zone difference, the plane takes off at 7:10 pm + 2 hours = 9:10 pm.

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The height of the white board for the given aspect ratio is found as 36 in.

Aspect ratio of a white board = 16:9 .

Width = 64 in

Let the height be 'h'.

So, using proportions.

16/9 = 64/h

h = 64*9 /16

h = 36

Thus, the height of the white board for the given aspect ratio is found as 36 in.

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Answer:

x = 40°

Step-by-step explanation:

given Δ ABD is isosceles with AB = AD

then the base angles are congruent , that is

∠ ABD = ∠ ADB = 25°

given ABC is a straight line then the 3 angles on it sum to 180° , that is

∠ ABD + x + 115° = 180°

25° + x + 115° = 180°

x + 140° = 180° ( subtract 140° from both sides )

x = 40°

For the scale factor of a dilation is 8: 7,

''This scale factor results in an enlarged image.''

A ratio indicates how many times one number contain in another number. The ratio of two number is written as x : y, which is equivalent to x/y.

Where, x and y are individual amount of two quantities.

And, Total quantity gives after combine as x + y.

Given that;

The scale factor of a dilation is 8: 7.

Now, We know that;

When the scale factor is between 0 and 1 the image is a reduction.

And, If the scale factor is greater than 1, the image is an enlargement.

Hence, For the scale factor of a dilation is 8: 7,

''This scale factor results in an enlarged image.''

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Answer:  x < -1

Step-by-step explanation:

-4x + 6 > 10

-4x > 10 - 6

-4x > 4

x < 4/-4

x < -1

Answer:

Step-by-step explanation:

[tex]\tt -4x + 6 > 10[/tex]

Subtract 6 from both sides:-

Divide both sides by -4:-

________________________

Hope this helps! :)

The simplified expression is a^(-(3)/(8)).

To simplify the expression (a^(-(1)/(2)))/(a^(-(1)/(8))), we need to use the properties of exponents. Specifically, we will use the property that states that when dividing two expressions with the same base, we can subtract the exponents.

So, in this case, we have:

(a^(-(1)/(2)))/(a^(-(1)/(8))) = a^(-(1)/(2) - (-(1)/(8)))

Simplifying the exponent:

= a^(-(4)/(8) + (1)/(8))

= a^(-(3)/(8))

In conclusion, the simplified expression of (a^(-(1)/(2)))/(a^(-(1)/(8))) is a^(-(3)/(8)).

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Answer:

The discount is 15% of the cost of the meal before the discount, so the discounted cost is:

Discounted cost = $78.97 - 0.15($78.97) = $67.12

Now we need to calculate the tip on the cost before the discount. The tip is 20% of the cost before the discount, so the tip amount is:

Tip = 0.20($78.97) = $15.79

Adding the discounted cost and the tip, we get the total cost of the meal:

Total cost = $67.12 + $15.79 = $82.91

Therefore, the total cost of the meal, including the discount and tip, is $82.91.

Based on the figure, the measure of the angle GPY is 118 degrees.

Alternate angles are a pair of angles that are formed by a transversal intersecting two parallel lines.

When a transversal intersects a parallel line, it creates four angles.

Alternate angles are the pairs of angles that are on opposite sides of the transversal and on opposite sides of the parallel lines.

The key property of alternate angles is that they are congruent, meaning they have the same measure.

Here angle GPY is alternate to angle RPK.

Thus;

8p + 94 = 5p + 103

8p - 5p = 103 - 94

3p = 9

Divide both sides by 3;

p = 3

The measure of angle GPY becomes ;

= 5(3) + 103= 15 + 103= 118 degrees.

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Answer:

24,000 grams

Step-by-step explanation:

<3

The values of x that satisfy the equation are x = -8 and x = 1.

To find all values of x that satisfy the equation x-(x+5)/(x+8)=(3)/(x+8), we can follow the steps below:

1. Multiply both sides of the equation by (x+8) to eliminate the fractions:
(x+8)(x) - (x+5) = 3

2. Distribute the (x+8) on the left side of the equation:
x^2 + 8x - x - 5 = 3

3. Combine like terms on the left side of the equation:
x^2 + 7x - 5 = 3

4. Subtract 3 from both sides of the equation to get the equation equal to zero:
x^2 + 7x - 8 = 0

5. Factor the left side of the equation:
(x+8)(x-1) = 0

6. Set each factor equal to zero and solve for x:
x+8 = 0 or x-1 = 0
x = -8 or x = 1

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The three possible forms of any positive odd integer, as required.

According to the Euclid Division Lemma, for any two positive integers a and b, there exist unique integers q and r such that a = bq + r where 0 ≤ r < b.

We can use this lemma to show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5 where q is some integer.

Let a be any positive odd integer and let b = 6. By the Euclid Division Lemma, we can write a = 6q + r where 0 ≤ r < 6. Since a is odd, r cannot be an even number. Therefore, r can only take on the values of 1, 3, or 5.

Thus, we can write a as:

a = 6q + 1
a = 6q + 3
a = 6q + 5

These are the three possible forms of any positive odd integer, as required.

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The gross profit margin of the cabinet is 50%.

Gross profit margin is the difference between the revenue and cost of an item divided by the revenue of the same item expressed as a percent.

Revenue refers to the amount a seller sells a particular units of product.

Cost refers to the amount spent by a producer to manufacture a particular unit of a product.

Revenue = $800

Cost = $400

Gross profit margin = (Revenue - Cost) / Revenue × 100

= ($800 - $400) / $800 × 100

= 400/800 × 100

= 50%

Ultimately, 50% is the gross profit margin of the item.

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Answer:

h is 19.5

Step-by-step explanation:

8.5 is 11 less than h

8.5 = h - 11

Add 11 to both sides

19.5 = h

Answer:-3

Step-by-step explanation:its just -9 + b/3 = -3 not even hard

There are a total of 32 possible sequences because there are two alternatives for each stage. Since there are more than seven distinct ways to mount the stairs, the priestess can do so every day for a week.

(a) Every day, the priestess at the temple of Horus must climb a flight of five stairs, which she can do by taking one or two steps at a time. The amount of ways she can up the stairs on any given day can be represented by a binary digit, where 1 denotes a two-step ascent and 0 denotes a one-step ascent. For each of the five steps, for instance, 00101 reflects the order of her one- and two-step climbs.

(b) The same regulations apply. The entire number of potential sequences, using the same binary digit format, is 213 = 8,192. There are therefore more than 365 possible ways to mount the grand staircase, allowing the priest to do so every day for a whole year.

In conclusion, the number of alternative binary sequences, where each digit denotes a one- or two-step climb, may be used to determine the number of ways the priestess can ascend the five-step flight of stairs and the number of ways the priest can ascend the thirteen-step grand staircase.

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Answer: The correct response is "Interest rates vary between lenders and can drastically change the monthly payment."

The interest rate is one of the most important things to consider when shopping around for loans because it determines the cost of borrowing money. Different lenders offer different interest rates, and even a small difference in interest rates can have a big impact on the total cost of the loan. A higher interest rate will result in a higher monthly payment, and a longer repayment period, which can make the loan more expensive overall. Therefore, it is important to shop around and compare interest rates from different lenders to find the best deal.

Step-by-step explanation:

Simplify both sides of the equation using algebraic methods and see if they are equal or not equal.

What is Linear Equation?

A linear equation is a mathematical equation that represents a straight line on a graph. It is an equation in which the highest power of the variable is one. A general linear equation can be represented as y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.

Linear equations can be solved algebraically using techniques like substitution or elimination, and they can be used to model real-world situations involving linear relationships.

If an equation is true for all values of x, then it is called an identity. To check if an equation is an identity, we can simplify both sides of the equation using algebraic methods and see if they are equal. If they are, then the equation is an identity.

On the other hand, if an equation is true for no values of x, then it is a contradiction. In this case, we can also simplify both sides of the equation using algebraic methods and see if they are not equal. If they are not, then the equation is a contradiction.

If neither of these cases applies, then the equation may be true for some values of x but not for others.

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The volume of the parallelepiped is 15.

To determine the volume of the parallelepiped with one vertex at the origin and the three vertices adjacent to it at (3,1,-1), (6,7,1), and (-6,8,9), we need to find the scalar triple product of the three vectors formed by these vertices. The scalar triple product is given by the determinant of the matrix formed by the three vectors:
\[ \begin{vmatrix} 3 & 1 & -1 \\ 6 & 7 & 1 \\ -6 & 8 & 9 \end{vmatrix} \]
Expanding the determinant, we get:
\[ \begin{aligned} \text { Volume } &=3\left( \begin{vmatrix} 7 & 1 \\ 8 & 9 \end{vmatrix} \right)-1\left( \begin{vmatrix} 6 & 1 \\ -6 & 9 \end{vmatrix} \right)-1\left( \begin{vmatrix} 6 & 7 \\ -6 & 8 \end{vmatrix} \right) \\ &=3(63-8)-1(54+6)-1(48+42) \\ &=165-60-90 \\ &=15 \end{aligned} \]
Therefore, the volume of the parallelepiped is 15.

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The correct name for the polynomial -2x^(3)+5x^(4)-3x is a quartic trinomial.

The student made an error when naming the polynomial as a "cubic trinomial". A cubic polynomial is one where the highest degree is 3, and a trinomial is a polynomial with three terms.

While this polynomial does have three terms, its highest degree is actually 4, as seen in the 5x^(4) term.

Therefore, the correct name for this polynomial would be a "quartic trinomial".

To summarize, the error the student made was naming the polynomial as a cubic trinomial instead of a quartic trinomial. The correct name for the polynomial -2x^(3)+5x^(4)-3x is a quartic trinomial.

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Answer: I need to woo

Step-by-step explanation: your exciting boy come find me

The lengths of a and b are 9.4 and 12 respectively using the Pythagorean theorem.

Pythagoras theorem states for a right angled triangle that, the sum of the squares of base and altitude is the square of the hypotenuse.

Given are two right angled triangles.

For the first right angled triangle, we have to find the length of the hypotenuse.

Using Pythagoras theorem,

a² = 8² + 5²

a² = 64 + 25

a² = 89

a = √89 = 9.43398 ≈ 9.4

For the second right angled triangle, we have to find the length of the altitude.

Using Pythagoras theorem,

17² = b² + 12²

b² = 17² - 12²

b² = 145

b = √145 = 12.04159 ≈ 12

Hence the length of and b are 9.4 and 12 respectively.

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Your question is incomplete. Most probably, the complete question with the image of the triangle is given below.

Work out the lengths of sides a and b.

Give your answers to 1 decimal place.

To find the composite functions, we simply substitute one function into the other.

For example, f[g(x)] means we substitute g(x) into f(x) wherever there is an x. So, f[g(x)] = 3(g(x)) + 4 = 3(2x^2 − 5x − 9) + 4 = 6x^2 − 15x − 23.Similarly, we can find the other composite functions: g[f(x)] = 2(f(x))^2 − 5(f(x)) − 9 = 2(3x+4)^2 − 5(3x+4) − 9 = 18x^2 + 7x - 7f[f(x)] = 3(f(x)) + 4 = 3(3x+4) + 4 = 9x + 16

For the last part, f/g means we divide f(x) by g(x):f/g = (3x+4)/(2x^2−5x−9)The domain of this function is all real numbers except for the values of x that make the denominator equal to zero.

To find these values, we set the denominator equal to zero and solve for x:2x^2−5x−9 = 0(2x+3)(x-3) = 0x = -3/2, 3 So the domain of f/g is (-∞, -3/2) ∪ (-3/2, 3) ∪ (3, ∞) in interval notation.

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