Find the critical points of f(x) = x - 18x² + 96x and use the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum. (Use symbolic notation and fractions when needed)

71.38 in word form is Seventy-one and thirty-eight hundredths.

What is the decimal number?

The accepted method for representing both integer and non-integer numbers is the decimal numeral system. Decimal notation is the term used to describe the method of representing numbers in the decimal system.

A number is a numerical unit of measurement and labeling in mathematics. The natural numbers 1, 2, 3, 4, and so on are the first examples. Number words are a linguistic way to express numbers.

To write decimals in word form, you can read the whole number part, write "and," and then read the decimal part according to place value.

For example: 0.25 can be written as "zero and twenty-five hundredths"

Hence, the correct answer is 'D. Seventy-one and thirty-eight hundredths'.

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A vertical translation (5 units up) is applied on quadratic function f(x) = x².

In this problem we find the representation of quadratic function and its image on Cartesian plane. The image is the consequence of using a vertical translation, whose definition is now introduced:

g(x) = f(x) + k

Where k is the y-coordinate of the quadratic function.

If we know that f(x) = x² and k = 5, then the image of the function is:

g(x) = x² + 5

The image is the result of a vertical translation (5 units up).

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To construct a confidence interval for the average bonus that all employees working for financial companies in New York City received last year, we need to know the sample size and the level of confidence. Without this information, we cannot calculate the confidence interval. Please provide the sample size and the level of confidence to proceed with the solution.

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In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

When Amelie divides the total number of muffins (385) by the number of muffins per box (9), she obtains a quotient of 42 and a remainder of 7. The quotient represents the number of complete boxes that Amelie can fill with 9 muffins each, while the remainder represents the number of muffins that cannot be put into a full box.

In other words, the quotient tells us how many full boxes Amelie can pack, and the remainder tells us how many muffins are left over after packing all the full boxes. In this case, Amelie can pack 42 full boxes, each with 9 muffins, which totals to 378 muffins. The remaining 7 muffins cannot fill a full box, and they are left over after all the full boxes are packed.

The "R 7" notation is commonly used to indicate the remainder in long division problems. The letter "R" stands for "remainder," and the number following it (in this case, 7) represents the actual remainder. The remainder is important because it indicates how many items are left over after dividing them into equal-sized groups.

In Amelie's situation, the remainder of 7 indicates that she has 7 muffins left over that cannot be packed into a full box of 9.

Overall, interpreting the "R 7" in this problem helps us understand how many full boxes of muffins Amelie can pack and how many muffins are left over after packing the full boxes.

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The vertices of triangle XYZ are (-4, 0), (1, -2), and (-2, 1).

We are given that X''(4, 0), Y''(2, -1), and Z''(2, 1). We can use these coordinates to determine the coordinates of the vertices of triangle XYZ.

Starting with X, we have (-y, x) = (4, 0). This implies that y = 0 and x = -4.

Moving on to Y we have (-z, y) = (2, -1). This implies that z = -2 and y = 1.

Finally, for Z, we have (-x, z) = (2, 1). This implies that x = -2 and z = 1.

Therefore, the vertices of triangle XYZ are (-4, 0), (1, -2), and (-2, 1).

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The total number of customers that the waiter had would be = 30 customers.

The total number of tables the waiter had = 5 tables

The total number of women at each table = 3

The total number of men at each table = 3

The total number of people one each table = 6

Therefore the total number of customers that the waiter attended to would be = 5×6 = 30

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To find the average amount that Sarah saved per month over the last six months, we need to add up the total amount saved and divide by the number of months.

Total amount saved = $135.00 + $144.00 + $104.00 + $80.00 + $90.00 + $160.00 = $713.00

Number of months = 6

Average amount saved per month = Total amount saved / Number of months = $713.00 / 6 = $118.83

Therefore, the correct answer is d. $118.83.

It is important to note that when working with numbers and calculations, accuracy is crucial. In this case, rounding off the answer to the nearest cent would result in a different answer.

Additionally, checking the calculations multiple times to ensure accuracy is always recommended.

Overall, tracking and analyzing expenses and savings is important for financial planning and achieving financial goals. By keeping track of how much she saved each month,

Sarah can make informed decisions about her spending and saving habits and adjust accordingly to reach her vacation savings goal.

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Brenton would have $330,027.55 more in his retirement savings had he started investing at age 25 instead of age 39, assuming monthly compounding and a 6% annual rate of return.

Brenton would have in his retirement savings if he started investing at age 25 instead of age 39, we need to calculate the future value of his investments in both scenarios and find the difference.

We'll use the formula for the future value of a series of equal payments (annuity) compounded monthly:

[tex]FV = P * (((1 + r)^nt - 1) / r)[/tex]

Where FV is the future value, P is the monthly payment ($250), r is the monthly interest rate (0.06 / 12), n is the number of times compounded per year (12), and t is the number of years.

Scenario 1 (investing since age 39):
t = 66 - 39 = 27 years

[tex]FV1 = 250 * (((1 + 0.06/12)^(12*27) - 1) / (0.06/12))[/tex]

FV1 ≈ $292,795.72

Scenario 2 (investing since age 25):

t = 66 - 25 = 41 years

[tex]FV2 = 250 * (((1 + 0.06/12)^(12*41) - 1) / (0.06/12))[/tex]

FV2 ≈ $622,823.27

Now, find the difference between the two scenarios:

Difference = FV2 - FV1

Difference ≈ $622,823.27 - $292,795.72

Difference ≈ $330,027.55

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Answer:

Principal = $21,592.31

Step-by-step explanation:

The formula for continuous compound interest is

[tex]A = Pe^r^t[/tex], where A is the amount (aka investment worth), r is the interest rate, and t is the time in years (the number e simply shows us that we're dealing with continuous compound interest)

Since we're already given that have A = $58,226.31, r = 0.062 (we must convert the percentage to a decimal by simply moving the decimal two places to the left, which is the same as dividing by 100), and t = 16 years, we can simply solve for P:

[tex]58226.31=Pe^(^0^.^0^6^2^*^1^6^)\\58226.31=Pe^0^.^9^9^2\\58226.31/(e^0^.^9^9^2)=P\\21592.31176=P\\21592.31=P[/tex]

The number of bricks that will be required to build the staircase is: 30 bricks

An arithmetic sequence is defined as one where you get the next term by adding a constant, called the common difference, to the previous term. A lot of formulas come from this simple fact. and they allow us to solve for any term in the sequence and even the sum of the first few terms.

The formula for the nth term of an arithmetic sequence is:

aₙ = a₁ + (n - 1)d

where:

a₁ is first term

d is common difference

n is nth term

We are given:

a₁ = 200

d = -4

aₙ = 84

Thus:

84 = 200 + (n - 1)(-4)

84 - 200 = -4n + 4

-4n = -120

n = 30

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The value of f(-2)  is 15.

Hence option D is correct.

The given expression is

[tex]\int\limits {f(x)} \, dx[/tex] = (1/3)x³ - 2x² +3x - 5

To find expression of f(x)

Differentiate it with respect to x

f(x) = x² - 4x + 3

Now put x = -2

f(-2) = (-2)² - 4(-2) + 3

       = 4 + 8 + 3

       = 15

Hence,

f(-2)  = 15

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The coordinate of the image point is (9,1).

There are eight types of rules for the transformation of a point. When the  takes place across a line, then the point (x,y) is changed to the point (y,x).

Given that the rule for the transformation of a point P(1,1) is [tex]R_{x=5} (P)[/tex], which defines the reflection of a point about a line, that is parallel to the y-axis. The line [tex]x=5[/tex] is like a mirror. So, the distance between the line and the image point is equal to the distance between the line and the original point.

Using the point-line distance formula, the distance between the line [tex]x=5[/tex] and a point (1,1) is given by [tex]|5-1|=4[/tex].

Similarly, by the above statement, the distance between the line [tex]x=5[/tex] and the image point will also be 4.

Therefore, the coordinate of the image point is (9,1).

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The complete question is -

If P = (1,1), then find the reflection [tex]R_{x=5} (P)[/tex].

Answer:

Hey!
You could obviously count how many you're taking, so that's 7 left behind.  My guess is that you could taste the jam... but that's the best I've got.

The requreid we can sneak out 9 cans of jam in the dark and still be sure that there will be at least 4 cans of one kind of jam and 3 cans of another left over.

It involves the basic operations of addition, subtraction, multiplication, and division, as well as more advanced operations such as exponents, roots, logarithms, and trigonometric functions.

Let's first find the minimum number of cans that need to be left in the cellar to meet the given criteria. We want at least 4 cans of one kind of jam and 3 cans of another leftover. This means we can take a maximum of:

8 - 4 = 4 cans of strawberry jam

7 - 3 = 4 cans of raspberry jam

5 - 3 = 2 cans of cherry jam

So, we can take a maximum of 4 + 4 + 2 = 10 cans in total.

To have certainty that we meet the criteria, we need to take one less than the maximum number of cans, which is 9 cans. So, we can sneak out 9 cans of jam in the dark and still be sure that there will be at least 4 cans of one kind of jam and 3 cans of another left over.

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The absolute maximum value on (0, ∞) for f(x) = 4x - 2x ln x is approximately 2e (5.436), which occurs at x = e.

To find the absolute maximum value on (0, ∞) for the function f(x) = 4x - 2x ln x, we need to follow these steps:
1. Determine the critical points of the function by finding its first derivative and setting it equal to zero.
2. Check the critical points for the maximum value.
3. Verify the behavior at the boundary of the interval (0, ∞).

Step 1: Find the first derivative of f(x).
f(x) = 4x - 2x ln x
f'(x) = d/dx (4x) - d/dx (2x ln x)
Using the product and constant rules, we get:
f'(x) = 4 - 2(ln x + 1)

Step 2: Set the first derivative equal to zero to find the critical points.
4 - 2(ln x + 1) = 0
2(ln x + 1) = 4
ln x + 1 = 2
ln x = 1
Solving for x:
x = e^1
x = e (approximately 2.718)

Step 3: Check the behavior at the boundaries.
As x approaches 0 from the right, ln x approaches negative infinity, making the term -2x ln x approach infinity. Since the interval is (0, ∞), we only need to consider the behavior as x approaches ∞. As x goes to infinity, both 4x and -2x ln x will also go to infinity. However, -2x ln x will increase at a slower rate compared to 4x, so f(x) will approach infinity.

Now, we need to check the value of f(x) at the critical point x = e:
f(e) = 4e - 2e ln e
f(e) = 4e - 2e(1)
f(e) = 2e (approximately 5.436)

Thus, the absolute maximum value on (0, ∞) for f(x) = 4x - 2x ln x is approximately 5.436, which occurs at x = e.

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Answer: 90 feet squared

Step-by-step explanation:

The formula for finding the area of a triangle is

(bxh)/2

In this problem, you need to convert 3 yards to feet. The conversion is as follows:

1 yard = 3 feet

Therefore, 3 yards = 9 feet.

You can then plug everything in and solve as follows:

20x9= 180

Then divide 180 by 2.

180/2=90

Answer:[tex]90ft^{2}[/tex]

Step-by-step explanation:

Area of a triangle: A=(bh)/2

b = 20 ft

h = 3 yds = 9ft

Plug in

A=[(20ft)(9ft)]/2

A=180ft/2

A=90ft^2

This circle with center C will be circumscribed around triangle △DEF, touching each vertex. The translated and dilated ⊙O will coincide with ⊙P, proving that they are similar.

1. To construct a circle with center C circumscribed about triangle △DEF, first find the perpendicular bisectors of all three sides of the triangle. Next, locate the point where the three perpendicular bisectors intersect, which will be the center C of the circumscribed circle. Finally, draw the circle using the distance from point C to any vertex of the triangle as the radius.

2. To prove that ⊙O and ⊙P are similar using similarity transformations on ⊙O, follow these steps:
  a. Translate ⊙O by the vector <-2+12, 7-(-1)> or <10, 8> so that its center coincides with the center of ⊙P. The translated ⊙O will have the center (10, 15).
  b. Since the ratio of the radii of ⊙O and ⊙P is 5:12, perform a dilation of ⊙O with a scale factor of 12/5 and center (10, 15). The dilated ⊙O will have a radius of 12, which is equal to the radius of ⊙P.
 

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The fair return for Castle Black Camping (CBC) is 20%.

This is calculated by subtracting the risk-free rate (2%) from the market's expected return (11%), and then multiplying the result by 2 (since CBC is twice as risky as the average stock).

To form a portfolio that is half Harlan County Safety Co. (HCSC) and half CBC, we need to calculate the portfolio's beta. This is done by multiplying each stock's beta by its weight in the portfolio, and then adding the results. In this case, the portfolio's beta would be 0.6 (1.2 x 0.5 + 2 x 0.5).

The fair return for the portfolio is then calculated by adding the risk-free rate to the product of the portfolio's beta and the market's expected return. This gives us a fair return of 16.4% for the HCSC and CBC portfolio.

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Answer:

x ~= 61.0°

Step-by-step explanation:

This is a trig question. The marked angle is unknown so we will use an inverse trig function to find a missing angle. On your calculator you may need to use the 2nd button or shift to get not just "cos" but the inverse cos. It might look like cos with a -1 exponent or "arccos" or "acos"

So the triangle is a right triangle. The angle is next to one of the given sides. "Next to" is called "adjacent" these mean the same thing. One side is adjacent to the angle. The other given side is the hypotenuse.

The trig function that puts together Adjacent and Hypotenuse is cosine.

cos (angle) = adj/hyp

We know the two sides. Fill them in.

cos (angle) = 31/64

We are looking for the angle.

cos x = 31/64

Use the inverse cos to find the angle.

cos^-1 (31/64) = x

This is a calculator activity. Unless your teacher has given you tables of values (this is how it was done before calculators) you have to use a calculator.

x = 61.0284677763

round to the nearest tenth.

x ~= 61.0

To construct a line parallel to a given line through a given point, there are two methods that can be used: the ruler and compass method or the parallel line equation method.

The ruler and compass method involves drawing a line through the given point that intersects the given line at a right angle. Then, using the compass, the distance between the given point and the intersection point is measured and transferred to a point on the given line. Finally, a line is drawn through the given point and the point on the given line to create a parallel line.

The parallel line equation method involves using the slope of the given line to find the slope of the parallel line. This is done by recognizing that parallel lines have the same slope. Then, using the point-slope equation of a line, the parallel line equation can be found by plugging in the given point and the calculated slope.

In summary, constructing a line parallel to a given line through a given point can be achieved using either the ruler and compass method or the parallel line equation method. Both methods are valid and can be used depending on personal preference and familiarity with the mathematical concepts involved.

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10 stereos must the salesperson sell to make that same amount of money under both plans.

Assume that in order for the salesperson to earn the same amount of money under both schemes, x stereos must be sold.

The salesperson will be paid a salary of $250 + $25 for each stereo sold under the first strategy. So, these are the total earnings:

First plan's total profits are $250 plus $25.

Instead of receiving a salary under the second proposal, the salesperson would be compensated with a $50 commission for each stereo sold. So, these are the total earnings:

Total income under the second plan equals $50x

We can set the two equations for total earnings to be equal in order to determine the value of x:

$250 + $25x = $50x

When we simplify the equation, we obtain:

$250 = $25x

x = 10

The sales person must thus sell 10 stereos in order to.

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Answer:

3.966m

Step-by-step explanation:

4m - (30cm + 40mm)

Converting cm and mm to metre by dividing by 100 and 1000 respectively

=> 4.000m - (30/100 m + 40/1000 m)

=> 4.000m - (0.030m + 0.004m)

=> 4.000m - 0.034m

=> 3.966m

Answer:

3.66m

Step-by-step explanation:

First, we have units measured in meters, centimeters, and millimeters.  This means we have to convert everything to the same measurement.

The easiest way is to convert everything to meters, as that's what the unit in the final answer will be.

To convert centimeters to meters, divide by 100

30/100=0.3

To convert millimeters to meters, divide by 1,000

40/1000=0.04

Next, plug the values back into the original equation:

4m-(0.3+0.04)

solve the parenthesis first

4-0.34

3.66

So, this equals 3.66 meters.

Hope this helps! :)

Kate's mistake was in Step 2: "Divide the volume of Wesley's tank by 2." Since she wants a tank with half the volume of Wesley's tank, she should have divided the volume of Wesley's tank by 2, not the radius.

The correct approach would be to use the formula for the volume of a sphere:

V = (4/3)πr^3

Since Wesley's tank has a diameter of 30 cm, its radius is 15 cm. Substituting this value into the formula gives:

V = (4/3)π(15)^3

V ≈ 14,137 cm^3

To find the radius of the smaller tank with half the volume, Kate should have solved for the radius using the formula for the volume of a sphere:

(4/3)πr^3 = (1/2)(4/3)π(15)^3

Simplifying this equation gives:

r^3 = (1/2)(15)^3

r ≈ 10.8 cm

Therefore, the approximate diameter of the smaller tank should be 21.6 cm (twice the radius), not 10.8 cm.

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The present ages of Raj and his daughter are respectively: 42 years and 12 years.

Present Age of Raj =y years

Present Age of daughter =x years

According to Question :

7 Years ago,

y − 7 = 7(x − 7)

⇒ 7x − y − 42 = 0.............(1)

And 3 Years from now

y + 3 = 3(x + 3)

⇒ 3x − y + 6 = 0............(2)

From eq (1) and eq (2)

Subtract eq 2 from eq 1 to get:

7x − 3x − y + y − 42 − 6 = 0

⇒ 4x = 48

⇒ x = 12

Putting x = 12 in Equation (2). we get,

(3 × 12) − y + 6 = 0

⇒ y = 42

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Answer:

£400

Step-by-step explanation:

first find 40% of £1000

40\100*1000

=400

Therefore answer is £400

A good description of the cross section shown that is parallel to the edge of the pyramid that measures 5​ millimeters is a triangle with base of 5 millimeters and height of 16 millimeters.

In Mathematics and Geometry, a square pyramid can be defined as a type of pyramid that has a square base, four (4) triangular sides, five (5) vertices, and eight (8) edges.

In Mathematics and Geometry, a triangle can be defined as a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

In this context, we can reasonably infer and logically deduce that the edge of the prism that measures 5 millimeters represents a triangle with base of 5 millimeters and height of 16 millimeters.

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You lose 515 dollars every time you lose 10 dollars.

Now that we know it takes 50 hours to lose 10 dollars, we can calculate how much money you lose every 10 dollars by

the hourly rate of 10.50 dollars by 50 hours and subtracting that amount from 10 dollars:

Money lost = (10.50 dollars/hour) x 50 hours - 10 dollars

Money lost = 525 dollars - 10 dollars

Money lost = 515 dollars

Therefore, you lose 515 dollars every time you lose 10 dollars.

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Mark and Diane's property tax is $3,231.80.

To calculate the property tax for Mark and Diane, we need to multiply the assessed value of their home by the tax rate.

First, we need to convert the tax rate from mills to a decimal. One mill is equal to one-tenth of one cent or 0.001 dollars. Therefore, 43.7 mills is equal to 0.0437 dollars.

Next, we can calculate the property tax as follows:

Property tax = Assessed value × Tax rate

Property tax = $74,000 × 0.0437

Property tax = $3,231.80

Therefore, Mark and Diane's property tax is $3,231.80.

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The common ratio for the geometric series 1, 1/2, 1/4, 1/8, ... is 1/2 and the sum of series is 2 which is finite.

To determine whether the series has an infinite sum, we can use the formula for the sum of an infinite geometric series, which is:

S = a/(1-r),

where a is the first term and r is the common ratio.

In this case, a = 1 and r = 1/2, so

S = 1/(1 - 1/2) = 2.

Since the value of S is finite and not infinite, we can conclude that the given geometric series has a finite sum of 2.

The common ratio of a geometric series is the ratio between consecutive terms. For example, in the series 1, 2, 4, 8, 16, ..., the common ratio is 2 because each term is obtained by multiplying the previous term by 2.

To determine whether a geometric series has an infinite sum, we can use the formula for the sum of an infinite geometric series, which is S = a/(1-r), where a is the first term and r is the common ratio.

If the value of r is between -1 and 1 (excluding -1), then the series has a finite sum. If the value of r is greater than 1 or less than -1, then the series has an infinite sum.

In the given series 1, 1/2, 1/4, 1/8, ..., the common ratio is 1/2. To find the sum of the series, we can use the formula S = a/(1-r) with a=1 and r=1/2, which gives S=2. Since the value of S is finite and not infinite, we can conclude that the given geometric series has a finite sum of 2.

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complete question:

Determine the common ratio for each of the following geometric series and determine which ones have an infinite sum 1,1/2,1/4,1/8....

The area of the region enclosed by one loop of the curve r = 4cos(θ) is 4 square units. The slope of the tangent line to the polar curve r=1 - 2sin(θ) at θ = π/4 is  2 + √2.

Area of region enclosed by one loop of the curve r = 4cos(2θ)

The curve r = 4cos(2θ) has two loops, and we need to find the area of one loop, which is from θ = 0 to θ = π/4.

To find the area, we use the formula for the area enclosed by a polar curve

A = (1/2) ∫[a,b] r^2 dθ

where r is the polar function, and a and b are the angles of the region we want to find the area for.

So, the area of one loop is

A = (1/2) ∫[0,π/4] (4cos(2θ))^2 dθ

= 8 ∫[0,π/4] cos^2(2θ) dθ

Using the identity cos(2θ) = (cos^2θ - sin^2θ), we can rewrite the integrand as

cos^2(2θ) = (cos^2θ - sin^2θ)^2

= cos^4θ - 2cos^2θsin^2θ + sin^4θ

= (1/2) (1 + cos(4θ)) - (1/2) sin^2(2θ)

So, the integral becomes

A = 8 ∫[0,π/4] [(1/2) (1 + cos(4θ)) - (1/2) sin^2(2θ)] dθ

= 4 [θ/2 + (1/8)sin(4θ) - (1/4)θ - (1/8)sin(2θ)]|[0,π/4]

= 1 + (2/π)

Therefore, the area of one loop of the curve r = 4cos(2θ) is 1 + (2/π).

Slope of tangent line to the polar curve r = 1-2sinθ at θ = π/4

To find the slope of the tangent line, we need to take the derivative of the polar function with respect to θ:

dr/dθ = -2cosθ

Then, we can use the formula for the slope of the tangent line in polar coordinates

dy/dx = (dy/dθ) / (dx/dθ) = (r sinθ) / (r cosθ) = tanθ + r dθ/dθ

At the point specified by θ = π/4, we have

r = 1 - 2sin(π/4) = 1 - √2/2 = (2 - √2)/2

dθ/dθ = 1

So, the slope of the tangent line is

dy/dx = tan(π/4) + r dθ/dθ

= 1 + (2 - √2)/2

= (4 + 2√2)/2

= 2 + √2

Therefore, the slope of the tangent line to the polar curve r = 1-2sinθ at θ = π/4 is 2 + √2.

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