the points on the curve [tex]y = 2x^3 + 3x^2 - 12x + 9 are (-2,29) \; and \; (1,2)[/tex]where the tangent line is horizontal.
Given :
The equation of the curve is [tex]y=2x^3\:+\:3x^2\:-\:12x\:+\:9[/tex]
Given tangent line is horizontal . Tangent line is horizontal when slope =0
Slope is nothing but the derivative.
So we find out x values where derivative =0
Lets take derivative for the given curve y
[tex]y=2x^3+3x^2-12x+9\\y'=2(3x^2)+3(2x)-12\\y'=6x^2+6x-12[/tex]
Now we set the derivative =0 and solve for
[tex]6x^2+6x-12=0\\Divide\; whole \; equation \; by \; 6\\x^2+x-2=0\\(x+2)(x-1)=0\\x+2=0, x=-2\\\\x-1=0, x=1[/tex]
So , the slope =0 when x=1 and x=-2
Now we find out the points . Use the original function
[tex]y=2x^3\:+\:3x^2\:-\:12x\:+\:9\\x=-2\\2\left(-2\right)^3+3\left(-2\right)^2-12\left(-2\right)+9=29\\(-2,29)\\\\x=1\\2\left(1\right)^3+3\left(1\right)^2-12\left(1\right)+9=2\\(1,2)[/tex]
the points on the curve [tex]y = 2x^3 + 3x^2 - 12x + 9 are (-2,29) \; and \; (1,2)[/tex]where the tangent line is horizontal.
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Jan saw 9 full moons in 252 how do I write this in a ratio
What is the area of this trapezoid? 96 in² 132 in² 168 in² 1344 in² Trapezoid A B C D with parallel sides D C and A B. Points F and E are between D and C. F E B A form a rectangle with 4 right angles. D F is 3 inches, F E is 8 inches., E C is 3 inches., E B is 12 inches., and A B is 8 inches.
Answer:
Option B. 132 inch²
Step-by-step explanation:
Since we know the formula to calculate the area of trapezoid is = [tex]\frac{1}{2}(height)(base)[/tex]
where height of the given trapezoid ABCD is side BE and two parallel sides are AB and DC.
It's given in the question
Side BE = 12 inches
AB = 8 inches
DC = (DF + FE + EC) = (3 + 8 + 3) = 14 inches
Now we put these values in the formula
Area of trapezoid ABCD = [tex]\frac{1}{2}(AB+CD).(BE) = \frac{1}{2}(8 + 14).(12)=\frac{1}{2}(22)(12)=(11).(12)=132[/tex]
Therefore answer is Option B. 132 inch²
Find the absolute maximum and minimum values of f on the set
d. f(x, y) = xy2 + 2, d = {(x, y) | x ≥ 0, y ≥ 0, x2 + y2 ≤ 3}
The absolute minimum and maximum values of a function are the least and highest values, the function can take.
The absolute maximum and minimum values of f are 4 and 2, respectively.
The function is given as:
[tex]\mathbf{f(x,y) = xy^2 + 2}[/tex]
First, we determine the partial derivatives of f(x,y)
[tex]\mathbf{f_x(x,y) = y^2}[/tex]
[tex]\mathbf{f_y(x,y) = 2xy}[/tex]
Substitute 0 for y in the first equation.
So, we have:
[tex]\mathbf{x = 0}[/tex]
This means that:
[tex]\mathbf{Critical\ Point = f(0,0) = 0}[/tex]
Next, we calculate the extreme values.
We have:
[tex]\mathbf{x^2 + y^2 \le 3}[/tex]
Express as an equation
[tex]\mathbf{x^2 + y^2 = 3}[/tex]
Make y the subject
[tex]\mathbf{y^2 = 3 - x^2}[/tex]
Take square roots
[tex]\mathbf{y = \sqrt{3 - x^2}}[/tex]
Substitute [tex]\mathbf{y = \sqrt{3 - x^2}}[/tex] in [tex]\mathbf{f(x,y) = xy^2 + 2}[/tex]
So, we have:
[tex]\mathbf{f(x) = x(3 -x^2) + 2}[/tex]
Open brackets
[tex]\mathbf{f(x) = 3x -x^3 + 2}[/tex]
Differentiate
[tex]\mathbf{f(x) = 3 -3x^2 }[/tex]
Set to 0
[tex]\mathbf{3 -3x^2 =0}[/tex]
Collect like terms
[tex]\mathbf{3x^2 =3}[/tex]
Divide through by 3
[tex]\mathbf{x^2 =1}[/tex]
Take square roots of both sides
[tex]\mathbf{x =\±\sqrt1}[/tex]
[tex]\mathbf{x =\±1}[/tex]
The domain is given as:
[tex]\mathbf{x \ge 1}[/tex]
So, the possible value of x is:
[tex]\mathbf{x =1}[/tex]
Substitute 1 for x in [tex]\mathbf{y = \sqrt{3 - x^2}}[/tex]
[tex]\mathbf{y = \sqrt{3 - 1^2}}[/tex]
[tex]\mathbf{y = \sqrt{3 - 1}}[/tex]
[tex]\mathbf{y = \sqrt{2}}[/tex]
So, we have:
[tex]\mathbf{(x,y) = (1,\sqrt{2})}[/tex]
This means that, the absolute maximum is at: [tex]\mathbf{(x,y) = (1,\sqrt{2})}[/tex], while the absolute minimum is at [tex]\mathbf{(x,y) = (0,0) }[/tex]
Substitute these values in f(x,y)
[tex]\mathbf{(x,y) = (0,0) }[/tex]
[tex]\mathbf{f(0,0) = 0 \times 0^2 + 2 = 2}[/tex]
[tex]\mathbf{(x,y) = (1,\sqrt{2})}[/tex]
[tex]\mathbf{f(1,\sqrt 2) = 1 \times (\sqrt 2)^2 + 2 = 4}[/tex]
Hence, the absolute maximum and minimum values of f are 4 and 2, respectively.
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The set of all points equidistant from the endpoints of a segment What term matches the definition? A) Central Angle Eliminate B) Perpendicular Bisector C) Segment D) Skew Lines
Add the sum of the numbers (−5.4) and (+8.2) to the opposite number of (−2 3/4 ).
Find the slope of the line through p and q. p(0, 0), q(10, 8)
To find the slope of a line passing through points P(0, 0) and Q(10, 8), use the formula (change in y) / (change in x). The slope is calculated as (8 - 0) / (10 - 0), which simplifies to 0.8.
Explanation:The question asks for the slope of the line that passes through two points, P(0, 0) and Q(10, 8). To find the slope, we use the formula:
Slope (m) = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)
Substituting the coordinates of points P and Q, we get:
m = (8 - 0) / (10 - 0) = 8 / 10 = 0.8
Therefore, the slope of the line through points P and Q is 0.8.
The ratio of the areas of two similar polygons can be found by using the ratio of their perimeters or the ratio of similarity and squaring it. Always Sometimes Never
The ratio of the areas of two similar polygons can be found by squaring the ratio of their perimeters or corresponding linear dimensions, but this method only works sometimes, specifically with similar polygons, and not with non-similar ones.
The ratio of the areas of two similar polygons is indeed found by taking the ratio of their corresponding linear dimensions and squaring it. This is because the area is a two-dimensional measurement, whereas perimeter and linear dimensions are one-dimensional. For similar polygons, which are polygons that have the same shape but may have different sizes, any linear dimensions (such as sides, perimeters, diagonals) are proportional.
When you have two similar polygons and you know the ratio of their perimeters, you can use this perimeter ratio to find the ratio of their corresponding side lengths since the ratio of the perimeters of similar polygons is equal to the ratio of their corresponding linear dimensions. After finding this linear ratio, you square it to obtain the ratio of their areas because if one dimension is multiplied by a scale factor 's', the area is multiplied by 's' squared (scale factor squared).
It is important to note, however, that this rule only applies to similar polygons (similarity) and not to polygons that aren't similar, since non-similar polygons do not have a constant ratio of corresponding linear dimensions. Therefore, to answer the original question, the method works sometimes, specifically when dealing with similar polygons.
solve this equation 12y=132
VWXY is a rhombus. find the measure
What is 0.125 Point rounded to the nearest hundredth
0.125 rounded to the nearest hundredth is 0.13.
We have,
To round to the nearest hundredth, we look at the digit in the thousandth place, which is 5.
Since 5 is greater than or equal to 5, we round up the hundredth digit (2) by increasing it by 1.
Therefore,
0.125 rounded to the nearest hundredth is 0.13.
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Mr,angelo sliced two pizzas into equal parts,what fraction greater than 1 names both pizzas
What is the greatest fraction you can make using the digits 4, 7, and 9?
Answer:
(7^9)/4 = 40,353,607/4
Step-by-step explanation:
Assuming each digit is used once and exponentiation is allowed, the largest numerator and smallest denominator will result in the largest fraction.
__
If other functions, such as factorial are allowed, then there might need to be a limit on the number of times they are applied. For example,
(7!)^(9!)/4 has about 1 million digits
something like ...
((7!)^(9!))!/4 has many more digits than that
and you can keep piling on the factorial symbols to any desired depth.
A marker is randomly selected from a drawer that contains 20 green, 44 orange, and 30 blue markers.Which statement is true?
P(green)≈0.21P(blue)≈0.41P(orange)≈0.
what is the probability of picking a vowel from the word invite?
Chi needs to simplify the expression below.
For a class gardening project, 1/3 of Mrs. Montoya's students planted marigolds and 1/2 planted tulips. What fraction of the class planted either marigolds or tulips?
PLEASE HELP WILL GIVE BRAINLIEST!!!!!!! This is the graph of f(x). What is the value of f(2)?
the question answers are A 2 B 16 C 8 D 4
A $65 coat is now on sale for $52. What percent discount is given?
The area of a rectangular barnyard is given by the trinomial 6x2 + 7x – 20. What are the possible dimensions of the barnyard? Use factoring. (1 point)
Answer:
c 2x+5 and 3x-4
Step-by-step explanation:
Suppose 6 of 15 students in a grade-school class develop influenza where 20 percent of grade-school students nationwide develope influenza.
1,600 principal earning 7%, compounded annually, after 33 years
Mark wants a new car that costs $30,000. He only has $500 in his savings account and $300 in his checking account. Which financing option should he choose?
Answer:
Mark should lease the car with 0% down payment.
Step-by-step explanation:
We know that the new car will cost = $30000.
Mark has only (500+300) = $800 with him.
That means Mark has only 2.66% of the cost of the car which is very less amount.
Thus Mark should have the car by leasing it with 0% down payment because in this way he will be able to buy or own the car with out giving any money for the car or without actually paying for the car and he can also at the same time can have his entire savings amount that he has as savings account and checking accounts.
Thus, he should lease the car with 0% down payment.
What expression represents "eight more than a number cubed and 2"?
Possible Answers:
What is the surface area of this right rectangular prism? Enter your answer in the box. in² A rectangular prism with a length of 3 inches, a width of 1 inch, and a height of 2 inches.
Answer:
Surface Area of Rectangular Prism is 22 [tex]in^2[/tex]
Step-by-step explanation:
Given: Dimensions of Rectangular prism are Length = 3 inches, width = 1 inch and height = 2 inches.
To find: Surface Area of Rectangular Prism
Formula Used for Surface Area of Rectangualr Prism
= 2 × ( Lenght × Width + Width × Height + Height × Length )
⇒ Surface Area = 2 × ( 3 × 1 + 1 × 2 + 2 × 3 )
= 2 × ( 3 + 2 + 6 )
= 2 × ( 11 )
= 22 [tex]in^2[/tex]
Therefore, Surface Area of Rectangular Prism is 22 [tex]in^2[/tex]
Answer:
22 inches
Step-by-step explanation:
20 PTS! What are the different shapes of a data set? Explain left and right skew and how this affects the mean of a data set.
Thanks, any help is much appreciated! :)
A grain of sand has a mass of 2.6 × 10–3 gram. A grain of salt has a mass of 6.5 × 10–2 gram. How many times greater is the mass of the grain of salt than the mass of the grain of sand?
Answer:
the person above me is correct
Step-by-step explanation:
Marissa bought 2 1/2 gallons of orange juice to make punch.Marissa pours 1 1/2 cups of punch in each glass.How many fluid ounces is each serving
Simplify. 7 to the second power + 3 (-9-7) + 88 divides by 8 + 3 to the second power
The number of miles that a commuter travels a car per day for one seven day work week is as follows: 19, 28, 17, 39, 22, 44, 56. The number of miles that a commuter travels on a bus per day for one seven day work week is as follows: 32, 26, 25, 40, 39, , 49, 22. Which method of transportation has a higher commuter average? Type either car or bus in the space provided. Use all lower case letters and do not use spaces in your answer.
Which of the following are equivalent to 85%? 0.085, 0.85, 85/100, 19/20 ? chose all that apply?