Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = 4x³ - 18x² - 48x + 7, [-2, 5]

Answer:

3

Step-by-step explanation:

first find g(-2) by plugging in -2 as x in the g(x) equation

g(-2) = 3(-2)+5

g(-2) = -6+5

g(-2) = -1

then plug that value (-1) as x in the f(x) equation

f(-1) = 4-(-1)^2

f(-1) = 4-1

f(-1) = 3

Answer:

-86,-85,-84,-83 and so on...

Step-by-step explanation:

Use the Number Line

Answer:

-88,-89,-90,-91,-92

Step-by-step explanation:

The solutions of the equation x² – 5x – 24 = 0 will be 8 and -3. Then the correct option is D.

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

The quadratic equation is given below.

x² – 5x – 24 = 0

Then the factor of the equation will be

       x² – 5x – 24 = 0

x² – 8x + 3x – 24 = 0

x(x – 8) + 3(x – 8) = 0

       (x – 8)(x + 3) = 0

                          x = 8, -3

Then the correct option is D.

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Answer:

  (a) No; rainfall amounts are positive

Step-by-step explanation:

Rainfall amounts are measured using a gauge that reports the amount as zero when the gauge is empty. Any rainfall adds to the amount in the gauge, so is reported as a positive number. There is no way to remove more rainfall than is in the gauge, so there is no way to have a negative rainfall amount

A prediction of a negative rainfall amount makes no sense, because you can't have a negative amount of rainfall.

According to scientific notation the decimal point is shifted to the right (option D).

Scientific notation is a term to refer to the way of writing numbers based on powers of 10. Generally this tool is used for very large or small values.

According to the above, the number 2,420 x 10⁵ expressed in decimal would look like this:

According to the above, it can be inferred that the point is placed in the final right part of the number, that is, it moves to the right.

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Answer:

A

Step-by-step explanation:

The equation of the line that passes through the points (-9,-7) and (-12,-2) is

y = -5/3x-22

and

equation of the line that passes through the points (1, 9) and (6, 6) is

y = -3/5x+48/5

clearly it is not the same line,

[tex]\frac{-5}{3}.\frac{-3}{5} \neq -1[/tex] (lines are not perpendicular)

[tex]\frac{-5}{3}\neq \frac{-3}{5}[/tex] (not parallel)

Answer: Option (3)

Step-by-step explanation:

The slope of the given line is

[tex]\frac{-2-4}{-6-2}=\frac{-6}{-8}=\frac{3}{4}[/tex]

Thus, the answer must be Option (3)

Answer:

Step-by-step explanation:

We can use the given interest rates and the investment and interest amounts to write an equation for the amount of interest Daran needs to earn.

Let b represent the amount Daran should invest in the bond fund (in thousands). Then 670 -x is the amount he will invest in the CD account. The total interest he wants to earn (in thousands) is ...

  0.085(b) +0.015(670 -b) = 40

Eliminating parentheses and collecting terms, we have ...

  0.07b +10.05 = 40

  0.07b = 29.95

  b = 427.857

Daran should invest $427,857 in the bond fund and $242,143 in the CD account.

Step-by-step explanation:

This triangle is acute! ✅

It is also scalene.

see the picture below

Answer:

• mean = 61

• mode = 50

• median = 57.5

Step-by-step explanation:

• The mean is calculated by adding all the values together, and dividing the result by the number of values.

∴ mean =  [tex]\frac{60 + 50 + 50 + 70 + 40 + 80 + 65 + 55 + 50 + 90}{10}[/tex]

⇒ mean = [tex]\frac{610}{10}[/tex]

⇒ mean = 61

• The mode of a set of values is the value that is the most common (has highest frequency) among them.

50 is the most common value.

∴ mode = 50

• The median is the middle-value of a set of ordered values.

∴ We have to first rearrange the set:

⇒  40, 50, 50, 50, 55, 60, 65, 70, 80, 90

Now we need to find the middlemost value:

Since we have 10 values, which is an even number, we have to use the formula:

median = [tex]\frac{(n/2)^{th} \space\ term \space\ + \space\ [(n/2) + 1]^{th} \space\ term }{2}[/tex]

where n is the number of values.

∴ median =  [tex]\frac{(10/2)^{th} \space\ term \space\ + \space\ [(10/2) + 1]^{th} \space\ term }{2}[/tex]

⇒ median =  [tex]\frac{5^{th} \space\ term \space\ + \space\ 6^{th} \space\ term }{2}[/tex]

The 5th and 6th terms in our ordered series are 55 and 60 respectively.

∴ median = [tex]\frac{55 + 60}{2}[/tex]

⇒ median = 57.5

Step-by-step explanation:

the expected value is calculated by multiplying each of the possible outcomes by their probability, and then summing up all these results.

tossing a coin 2 times gives us 4 possible different outcomes (all with the same probability of 0.25) :

head - tail

head - head

tail - head

tail - tail

to have exactly one head is 2 out of these 4 possible outcomes, and the probability is 0.5.

everything else is also 2 out of these 4 possible outcomes, and the probability for that is therefore 0.5 too.

the expected value (from your point of view) is

9×0.5 - 15×0.5 = -$3.00

Compute the gradient of [tex]f[/tex].

[tex]\nabla f(x,y) = \left\langle 3x^2 - 6y, -6x + 3y^2\right\rangle[/tex]

Set this equal to the zero vector and solve for the critical points.

[tex]3x^2-6y = 0 \implies x^2 = 2y[/tex]

[tex]-6x+3y^2=0 \implies y^2 = 2x \implies y = \pm\sqrt{2x}[/tex]

[tex]\implies x^2 = \pm2\sqrt{2x}[/tex]

[tex]\implies x^4 = 8x[/tex]

[tex]\implies x^4 - 8x = 0[/tex]

[tex]\implies x (x-2) (x^2 + 2x + 4) = 0[/tex]

[tex]\implies x = 0 \text{ or } x-2 = 0 \text{ or } x^2 + 2x + 4 = 0[/tex]

[tex]\implies x = 0 \text{ or } x = 2 \text{ or } (x+1)^2 + 3 = 0[/tex]

The last case has no real solution, so we can ignore it.

Now,

[tex]x=0 \implies 0^2 = 2y \implies y=0[/tex]

[tex]x=2 \implies 2^2 = 2y \implies y=2[/tex]

so we have two critical points (0, 0) and (2, 2).

Compute the Hessian matrix (i.e. Jacobian of the gradient).

[tex]H(x,y) = \begin{bmatrix} 6x & -6 \\ -6 & 6y \end{bmatrix}[/tex]

Check the sign of the determinant of the Hessian at each of the critical points.

[tex]\det H(0,0) = \begin{vmatrix} 0 & -6 \\ -6 & 0 \end{vmatrix} = -36 < 0[/tex]

which indicates a saddle point at (0, 0);

[tex]\det H(2,2) = \begin{vmatrix} 12 & -6 \\ -6 & 12 \end{vmatrix} = 108 > 0[/tex]

We also have [tex]f_{xx}(2,2) = 12 > 0[/tex], which together indicate a local minimum at (2, 2).

Answer:

Step-by-step explanation:

One number is 2 more than another. The difference between their squares is 24. What are the numbers?

(x + 2)² - x² = 24

x² + 4x + 4 - x² = 24

4x + 4 = 24

4x = 20

x = 20 : 4

x = 5

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

5 + 2 = 7

check

(5 + 2)² - 5² = 24

49 - 25 = 24

24 = 24

the answer is good

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Answer:  [tex]\textsf{285 total calories}[/tex]

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Given: [tex]\textsf{1 egg = 95 calories}[/tex]

Find: [tex]\textsf{Calories in 3 eggs}[/tex]

Solution: In order to determine the total amount of calories in 3 eggs we must multiply the amount of calories in one egg by 3.  So we do 95 * 3 which gives us 285 total calories.

It really depends what situation you are trying to set goals at,

Trying to understand math concept or any really concept, I would set little goals such as " Study for 15 mins a day" or something and the big goal would be "UNDERSTAND THE CONCEPT"

for goals, I would set small goals that work towards your main goal.

Answer:

Marriage and Family Harmony. ...

Proper Mindset and Balance. ...

Commitment to Improved Physical Health. ...

Career Passion and Personal Satisfaction. ...

Develop Empathy and Gentleness. ...

Financial Stability. ...

Service and Social Responsibility.

The probability that the family has two dogs or two cats will be 0.5.

From the information, the head represents cats and rte tails represent dogs.

The experimental probability that the family has two dogs or two cats will be:

= (1/2 × 1/2) + (1/2 × 1/2)

= 1/4 + 1/4

= 1/2

= 0.5

Also, the theoretical probability that they have three dogs or three cats is 0.5.

The simulation to generate data for three pets will be to add a new coin and category for the new pet.

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The  rule that describes the composition of transformations that maps ΔJKL to ΔJ"K"L" is Rotation  0° to  90°  and reflection about the x-axis.

A transformation is a rule that is used to manipulate the position of a point of geometric figure.

Analyzing the figure, rotation of ΔJKL through the angle 90 degrees in a counter-clockwise direction gives us  ΔJ'K'L' .

ΔJ"K"L" is been gotten also using ΔJ'K'L'  through the refraction of ΔJ'K'L' across the x-axis.

In this case,   rule that describes the composition of transformations that maps ΔJKL to ΔJ"K"L" is Rotation  0° to  90°  and reflection about the x-axis.

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Answer: C). 90 degree rotation about point 0 composition translation of negative 2 units x, 0 units y

Step-by-step explanation:

The time 0.825 is 49.5 minutes based on the conclusion that the sum of the different amounts of time was 7.825 hours.

Proportion in time refers to the relationship between two or more ratios or quantities expressed in time measurements.

One hour = 60 minutes

One minute = 60 seconds

0.825 of an hour = 49.5 minutes (0.825 x 60 minutes)

0.5 minutes = 30 seconds (0.5 x 60)

Thus, we can conclude that the total time involved is 7 hours, 49 minutes, and 30 seconds.

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1) If the limit [tex]L[/tex] is

[tex]L = \displaystyle \lim_{\Delta x\to0} \frac{\sin\left(\frac\pi3 + \Delta x\right) - \sin\left(\frac\pi3\right)}{\Delta x}[/tex]

then using the hint as well as [tex]\sin\left(\frac\pi3\right)=\frac{\sqrt3}2[/tex] and [tex]\cos\left(\frac\pi3\right)=\frac12[/tex] we have

[tex]\displaystyle L = \lim_{\Delta x\to0} \frac{\sin\left(\frac\pi3\right)\cos(\Delta x) + \cos\left(\frac\pi3\right)\sin(\Delta x) - \sin\left(\frac\pi3\right)}{\Delta x}[/tex]

[tex]\displaystyle L = \sin\left(\frac\pi3\right) \lim_{\Delta x\to0} \frac{\cos(\Delta x) - 1}{\Delta x} + \cos\left(\frac\pi3\right) \lim_{\Delta x} \frac{\sin(\Delta x)}{\Delta x}[/tex]

[tex]L = \dfrac{\sqrt3}2 \times 0 + \dfrac12\times1 = \boxed{\dfrac12}[/tex]

which follows from the well-known limits,

[tex]\displaystyle \lim_{x\to0} \frac{1-\cos(x)}x = 0 \text{ and } \lim_{x\to0} \frac{\sin(x)}x=1[/tex]

Alternatively, if you already know about derivatives, we can identify the limit as the derivative of [tex]\sin(x)[/tex] at [tex]x=\frac\pi3[/tex], which is [tex]\cos\left(\frac\pi3\right)=\frac12[/tex].

2) It looks like you may be using double square brackets deliberately to denote the greatest integer or floor function which rounds the input down to the nearest integer. That is, [tex][\![x]\!][/tex] is the greatest integer that is less than or equal to [tex]x[/tex]. The existence of [tex]L[/tex] depends on the equality of the one-sided limits.

Suppose [tex]3\le x<4[/tex]. Then [tex]2[tex]\displaystyle \lim_{x\to4^-} [\![x - 1]\!] = 2[/tex]

Now suppose [tex]4\le x<5[/tex], so that [tex]3\le x-1 < 4 \implies [\![x-1]\!]=3[/tex] and

[tex]\displaystyle \lim_{x\to4^+} [\![x-1]\!] = 3[/tex]

The one-sided limits don't match so the limit doesn't exist.

Answer:

help to others the exact amount u have

Step-by-step explanation:

believe it or not, the area of any triangle is

baseline × height / 2

we have here 2 triangles, both with the same baseline (8 cm), but with 2 different heights : 6.5 cm and 4.6 cm.

so, their areas are

8 × 6.5 / 2 = 4 × 6.5 = 26 cm²

8 × 4.6 / 2 = 4 × 4.6 = 18.4 cm²

so, in total, the shaded region is

26 + 18.4 = 44.4 cm²

Answer:

6 hours

Step-by-step explanation:

Let the smaller pipe fill the pool in x hours and the bigger pipe y hours.

x-y=4

1/x+1/y=4/15

x=10, y=6

so 6 hours

Step-by-step explanation:

l = large pipe

s = small pipe

s needs x hours alone to fill the pool.

l needs x-4 hours alone to fill the pool.

s does 1/x of the work in 1 hour.

l does 1/(x-4) of the work in 1 hour.

3 hours 45 minutes = 3.75 hours

3.75 × 1/x + 3.75 × 1/(x-4) = 1 (= the whole work)

3.75 + 3.75 × x/(x-4) = x

3.75×(x-4) + 3.75×x = x(x-4) = x² - 4x

3.75x - 15 + 3.75x = x² - 4x

7.5x - 15 = x² - 4x

-15 = x² - 11.5x

0 = x² - 11.5x + 15

the general solution to such a quadratic equation is

x = (-b ± sqrt(b² - 4ac))/(2a)

in our case

a = 1

b = -11.5

c = 15

x = (11.5 ± sqrt(132.25 - 4×1×15))/(2×1) =

= (11.5 ± sqrt(72.25))/2 = (11.5 ± 8.5)/2

x1 = (11.5 + 8.5)/2 = 20/2 = 10

x2 = (11.5 - 8.5)/2 = 3/2 = 1.5

x = 1.5 would turn l = 1/(x-4) negative, which does not make sense.

so, x = 10 is our solution.

that means

s does 1/10 of the work in 1 hour.

so, the small pipe alone fills the pool in 10 hours.

l does 1/(10-4) = 1/6 of the work in 1 hour.

so, the large pipe alone fills the pool in 6 hours.

Step-by-step explanation:

The derivative of log is

[tex] \frac{d}{dx} ( log_{a}(x) ) = \frac{1}{x \: ln(a) } [/tex]

You can easily derive this using the Change of base rule, and natural log rules

[tex] log_{a}(x) = \frac{ log_{e}(x) }{ log_{e}(a) } = \frac{ ln(x) }{ ln(a) } [/tex]

Next, we differentiate with respect to x.

[tex] \frac{d}{dx} \frac{ ln(x) }{ ln(a) } = \frac{1}{ ln(a) } \times \frac{d}{dx} ln(x) [/tex]

[tex] = \frac{1}{x ln(a) } [/tex]

So back to the topic at hand,.

a=7, so we get

[tex] \frac{1}{x ln(7) } [/tex]

We plug in 0, we would get undefined

Answer:

x=1/3

Step-by-step explanation:

-4(3x-2)=6x+2

or,-12x+8=6x+2

or,-12x-6x=2-8

or,-18x=-6

or,x=-6/-18

or,x=1/3

Answer:

Step-by-step explanation:

The area of a parallelogram is h*b where h is the height/altitude and b is the base.

The base is 27 millimeters and the altitude is 18 millimeters, so we multiply those together.

27*18= 486.

The area of the parallelogram is 486mm²

Hope this helps!

Answer:

C. no, because the arcs do not have the same measure.

Step-by-step explanation:

for 2 structures to be congruent all the angles need to be the same (and then all the lines must have the same scaling factor between both structures - but that is not relevant here anymore, since the angles are already off).

The weight of relish prepared after the evaporation of water is 2,226.02 pounds

Total = 1000 + 500 + 900 + 75 + 5

= 2,480 pounds

pickles are 92% water

= 92/100 × 1000

= 920 pounds

onions are 85% water

= 85/100 × 500

= 425 pounds

salt brine is 88% water

= 88/100 × 75 pounds

= 66 pounds

Total water in the mixture = 920 pounds + 425 pounds + 66 pounds

= 1,411 pounds

Amount of water evaporated = 18/100 × 1,411 pounds

= 253.98 pounds

Weight of the relish prepared = 2,480 pounds - 253.98 pounds

= 2,226.02 pounds

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The area of the shaded face of the cylinder with the given radius is approximately 1520cm².

This question is incomplete, the missing diagram is uploaded along the answer below.

From the diagram, the shaded region is a circle with radius 22 meters.

Area of a circle is expressed as;

A = π × r²

r is radius and π is constant pi ( π = 3.14 )

We substitute our values into the equation above.

A = π × r²

A = 3.14 × (22m)²

A = 3.14 × 484m²

A = 1519.76 ≈ 1520cm²

Therefore, the area of the shaded face of the cylinder with the given radius is approximately 1520cm².

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Answer:

4x+2y=4

3x-y=-7

first one by 3

second by 4

12x+6y=12

12x-4y=-28

subtract

10y=40

y=4

plug in

3x-4=-7

add 4

3x=-3

x=-1

(-1,4)

c

Hope This Helps!!!

Answer: third option

Step-by-step explanation:

There are 2 ways to go about systems of equations normally, and that's elimination and substitution. For this particular problem, I would recommend elimination.

we see the first equation 4x+2y=4

we want to be able to simplify it as much as possible (so it's easier to solve)

so we divide all the terms by 2, giving 2x+y=2.

the second equation can't be simplified, so we set up the elimination method.

   2x+y=2

  -3x-y=-7

multiplying both equations by 2 and 3, we subtract them and get that y equals 4, which is clearly the answer.

Answer:

145,152

Step-by-step explanation:

So there's actually two things you'll need in this equation. You'll need to use pascals triangle, to find the coefficients, and the binomial theorem to find the degrees.

So in this case you want to find the 9th row of pascals triangle. You could write out the entire 10 rows of pascals triangle to find the 9th row (because the first row is row 0). So to find the nth row you generally will start with 1 as the first term as every single row will start with this number. Now after that you're going to continue "row" amount of times starting with 2 values. One value which I'll name k=0 and the other j=1

Now let's start with what we have {1}, take the previous term and (which will always be 1 in the first case, and then multiply it by (row - k) / j. So in this case k=0, and j=1, row=9. You'll have (1 * (9)) / 1. This gives you 9, which is the second term. So now you have the terms {1, 9}. Now the next iteration add 1 to k and 1 to j. And the previous is now 9. So now you have (9 * (9 - 1)) / 2 = (9 * 8) / 2 = 72 / 2 = 36. Which is the next term. {1, 9, 36}. and then continue this until you have row+1 amount of numbers, or in other words repeat it "row" amount of times, since you already started with 1.

In doing so you will get {1, 9, 36, 84, 126, 126, 84, 36, 9, 1}

Now to use binomial theorem to expand find the degrees. Cause currently we can determine the coefficients, but what about the degrees? So it's as if we have the equation: [tex]1(2x)^a(3)^b+9(2x)^c(3)^d+36(2x)^e(3)^f.....[/tex] and so on until we use up all the numbers in the row. But binomial theorem essentially states that the degrees will expand as such: [tex]C_0a^nb^0+C_1+a^{n-1}b^{0+1}+C_2a^{n-2}b^{0+2}...C_na^{n-n}b^{n}[/tex]. So essentially the degree of a starts at n (in this case 9), and and then from there continues to go down until it reaches 0, while the degree of b starts at 0 (so it's just 1) and continues to go up (by 1) until it reaches n

So expanding it out gives us:

[tex]1(2x)^9(3)^0 + 9(2x)^{9-1}(3)^{0+1}+36(2x)^{9-2}(3)^{0+2}+84(2x)^{9-3}(3)^{0+3}[/tex]... and so on But in this case we really only care about degree six which occurs at 9-3. So let's focus on that

[tex]84(2x)^6(3)^3[/tex]. This evaluates out to [tex]84(64x^6)(27)[/tex] which then evaluates to [tex]145,152x^6[/tex]