The function f(x, y) = x³ + y³ - 3x² - 9y² - 9x has local maximum values at (-3, 0) and (1, 0), and a saddle point at (0, 3).
To find the critical points, we need to find the values of x and y where the partial derivatives of f with respect to x and y are equal to zero. Taking the partial derivatives, we get:
∂f/∂x = 3x² - 6x - 9 = 0
∂f/∂y = 3y² - 18y = 0
Solving these equations, we find the critical points to be (x, y) = (-3, 0), (1, 0), and (0, 3).
To determine the nature of these critical points, we can use the second partial derivative test. Computing the second partial derivatives:
∂²f/∂x² = 6x - 6
∂²f/∂y² = 6y - 18
∂²f/∂x∂y = 0
Substituting the critical points into the second partial derivatives, we find that:
∂²f/∂x²(-3, 0) = -24
∂²f/∂x²(1, 0) = -6
∂²f/∂x²(0, 3) = 0
Based on the sign of the second partial derivatives, we can determine the nature of each critical point. The point (-3, 0) has a negative second derivative, indicating a local maximum. The point (1, 0) has a negative second derivative, indicating a local maximum as well. Finally, the point (0, 3) has a second derivative equal to zero, indicating a saddle point.
Therefore, the function has local maximum values at (-3, 0) and (1, 0), and a saddle point at (0, 3).
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All the events in the sample space that are not part of the specified event are called _____. Group of answer choices joint events the complement of the event simple events independent events
All the events in the sample space that are not part of the specified event are called Complement of Event.
Complement of event:
The complement of an event is the subset of outcomes in the sample space that are not in the event. A complement is itself an event. The complement of an event A is denoted as A c A^c Ac or A′. An event and its complement are mutually exclusive and exhaustive.
Two events are said to be complementary when one event occurs if and only if the other does not. The probabilities of two complimentary events add up to 1. For example, rolling a 5 or greater and rolling a 4 or less on a die are complementary events, because a roll is 5 or greater if and only if it is not 4 or less.
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Which two statements would produce a tautology? (p q) (p q) (p q) (p q) q (p q) (p q) p
(p p) would produce Tautology.
Tautology:
It is needless repetition of an idea, statement, or word Rhetorical repetition, tautology ('always and for ever'), banal metaphor, and short paragraphs are part of the jargon.— Philip Howard. b : an instance of such repetition The phrase "a beginner who has just started" is a tautology.
A tautology is an expression or phrase that says the same thing twice, just in a different way. For this reason, tautology is usually undesirable, as it can make you sound wordier than you need to be and make you appear foolish.
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The sixth grade band students are selling magazines to raise money for new instruments. As a reward for selling, the students earn points to spend on prizes. Each student gets 25 points to start with, and then they earn an additional 5 points for each magazine they sell. If m represents the number of magazines a student sells, write an expression to represent the total number of points the student has.
WRITER
Answer:
okay he's saying that each student gets 25 points to start with and then they get five more like 25 / 25 * 5 or like 25 + 5 or 25 - 5 like represent those numbers and like get the number or something like that
Question 14 of 23
Which of the following describes a compound event?
OA. spinning a 3 on a spinner
OB. rolling a 3 on a die
OC. getting tails on a coin toss
OD. rolling a 5 on a die and drawing a 10 from a deck of cards
SUBMIT
Answer:
B
Step-by-step explanation:
ABC Corporation recently issued an IPO. Shortly thereafter it decides it wants to buy back some of its shares. What time frame restrictions would ABC face in order to buy back their own stock
The time restriction faced by ABC Corporation for it to buy back its newly issued shares is 4 weeks
What is share buyback?
Share buyback means that the company would repurchase its own shares in the stock market by paying investors holding the shares, then reduce its number of shares outstanding by the number of shares bought back.
The shares repurchased are known as Treasury stocks, which are held until the company decides to reissue them or cancel them.
The restriction placed on a company before it can repurchase its shares is that the shares must be in issue 4 weeks, in other words, the new issued shares through Initial Public Offer(IPO) must be existence for 4 weeks before being repurchased by the company after having issued an IPO.
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Find the measure of each numbered angle.
Answer: [tex]m\angle 1=82^{\circ}, m\angle 2=128^{\circ}[/tex]
Step-by-step explanation:
By the same-side interior angles theorem,
[tex]m\angle 1=180^{\circ}-98^{\circ}=82^{\circ}\\\\m\angle 2=180^{\circ}-52^{\circ}=128^{\circ}[/tex]
8.8÷4=8, point, 8, divided by, 4, equals
\text{ tenths}\div 4 tenths÷4start text, space, t, e, n, t, h, s, end text, divided by, 4
8.8 \div 4 =8.8÷4=8, point, 8, divided by, 4, equals
\text{ tenths} tenthsstart text, space, t, e, n, t, h, s, end text
8.8 \div 4 =8.8÷4=8, point, 8, divided by, 4, equals
The results of the given task 8.8 ÷ 4 is 2.2.
Division of Decimal8.8 ÷ 4
= 2.2
Therefore, 8.8 ÷ 4 = 2.2
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The coordinates of a quadrilateral are (2,1), (-1,3), (-5,-3), and (-2,-5).
The quadrilateral is a because are parallel and since the product of the slopes of both pairs of segments is -1,
The quadrilateral is a parallelogram because both pairs of opposite sides are parallel and since the product of the slopes of both pairs of adjacent segments is -1, the quadrilateral is a rectangle
Answer:
The guy above is somewhat right.
Hope this helps!
Step-by-step explanation:
The quadrilateral is a rectangle.
You have learned about the six trigonometric functions, their definitions, how to use them, and how to represent them graphically. The sine, cosine, and tangent trigonometric functions can be paired with their reciprocal functions, cosecant, secant, and cotangent, respectively. Think about how each function is related to its reciprocal function.
How are the graphs of the reciprocal functions related to their corresponding original functions? What happens to the graphs of the reciprocal functions as x approaches the zeros of the original functions? Describe how you would teach friends with different learning styles (visual-spatial, aural-auditory, verbal-linguistic, physical-bodily-kinesthetic, logical-mathematical, social-interpersonal, and solitary-intrapersonal) how to graph the reciprocal functions.
The graphs of the reciprocal functions are related to their corresponding original functions based on the information illustrated below.
What are trigonometric functions?It should be noted that the trigonometric functions are the real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
There are six functions of an angle. They are sine (sin), cosine (cos), tangent (tan), cotangent, secant, and cosecant (csc).
It should be noted that all the y-coordinates of a reciprocal function will be the reciprocals of the y-coordinates of the original function. Then, the graph of a reciprocal function has a vertical asymptote at each zero of the original function. For example, y=arcsinx reflects against y=sinx.
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The density of a certain material is such that it weighs 5 ounces per gallon of volume what is the density in pounds per cubic meter
The density in pounds per cubic meter is 83 pounds per cubic metre.
How to find density?The density is mass of a unit volume of a material substance.
In other words, density is the ratio between mass and volume or mass per unit .
Therefore, density can be represented mathematically as follows:
density = mass / volume
Hence,
mass = 5 ounces
5 ounces = 0.3125 pounds
1 gallon = 0.00378541 pounds
Therefore,
density = 0.3125 / 0.00378541
density = 82.5538052681
density = 83 pounds per cubic metre.
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A square has a side length of 10 inches. Congruent isosceles right triangles are cut off each corner so that the resulting octagon has equal side lengths. How many inches are in the length of one side of the octagon
The length of the side of the octagon is 4.142 inches.
We can find the length of the sides of the octagon as follows:Let the length of the congruent sides of the isosceles triangle be x.
The hypotenuse of the triangle is given by:
√x+x = √2x
This is also the length of the sides of the octagon. It is given that the sides of the octagon are equal.
It is also given that the length of the square side is 10 inches.
This means that:
x + √2x + x = 10
2x + √2x = 10
(2 + √2)x = 10
(2 + 1.414)x = 10
3.414x = 10
x = 10/3.414
x = 2.929 inches.
The side of the octagon is given by:
√2 × 2.929 = 4.142 inches
Therefore, we have found that the length of the sides of the octagon is 4.142 inches.
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What are the first five terms in the
recursive sequence defined by the
following?
The first five terms of the sequence are {0, -1.4, -2.8, -4.2, -5.6}
Recursive functionGiven the nth term of a recursive expression shown below
an =an-1 - 1.4
where
an-1 is the preceding term
a1 is the first term
an is the nth term
an-1 is
Given the following
a1 = 0
For the second term a2
a2 = 0 - 1.4
a2 = -1.4
For the third term a3
a3 = -1.4 - 1.4
a3 = -2.8
For the fourth term a4
a4 = -2.8 - 1.4
a4 = -4.2
For the fifth term a2
a5 = -4.2 - 1.4
a5 = -5.6
Hence the first five terms of the sequence are {0, -1.4, -2.8, -4.2, -5.6}
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I forgot how to do this, I need an explanation to find the answer
Answer:
23.53
Step-by-step explanation:
OK so we got sin 52°=0.8 and the cos 52°=0.6
but we need tan 52° because we don't want to spend our time on calculating hypothenuse so
[tex] \frac{ \sin(52) }{ \cos(52) } = 1.3 = \tan(52) [/tex]
so
[tex]1.3 = \frac{x}{18.1} = = = > x = 23.53[/tex]
help me pleaseeeeeeee
Answer:
x = -4/3 or x = 3/2
Step-by-step explanation:
Let's solve your equation step-by-step.
6x^2−x−12=0
Step 1: Factor left side of equation.
(3x+4)(2x−3)=0
Step 2: Set factors equal to 0.
3x+4=0 or 2x−3=0
x = -4/3 or x = 3/2
Answer:
x = -4/3 or x = 3/2
If the price of gas started at $2.56 per gallon and increased at a rate of 4% per year, after how many years will the price of gasoline per gallon reach or exceed $5?
The price of gas is modeled by f(x) = 2.56(1.04)x.
Use logarithms to find the answer to the question.
Answer: 18 years
Work Shown:
[tex]f(\text{x}) = 2.56(1.04)^\text{x}\\\\5 = 2.56(1.04)^\text{x}\\\\5/2.56 = (1.04)^\text{x}\\\\1.953125 = (1.04)^\text{x}\\\\\log(1.953125) = \log(1.04^\text{x})\\\\\log(1.953125) = \text{x}*\log(1.04)\\\\\text{x} = \frac{\log(1.953125)}{\log(1.04)}\\\\\text{x} \approx 17.0682937693249\\\\[/tex]
The steps above show f(x) replaced with 5. Then you'd use logarithms to isolate the variable x. The relevant useful log rule is [tex]\log(A^B) = B\log(A)[/tex] so we can pull down the exponent.
From here it seems your teacher wants you to round up to the nearest integer.
If we plugged in x = 17, then f(x) = 4.99 which is one cent too small.
While x = 18 leads to f(x) = 5.19
Therefore, x = 18 has the price reach or exceed $5
To fit in an existing frame, the length, x, of a piece of glass must be longer than 12 cm but not longer than 12.2 cm. Which inequality can be used to represent the lengths of the glass that will fit in the frame?
12 < x ≤ 12.2
12 > x ≤ 12.2
x > 12 or x ≤ 12.2
x < 12 or x ≤ 12.2
Considering the definition of an inequality, the inequality that expresses the previous relationship is 12 < x ≤ 12.2
Definition of inequalityAn inequality is the existing inequality between two algebraic expressions, connected through the signs:
greater than >.less than <.less than or equal to ≤.greater than or equal to ≥.An inequality contains one or more unknown values called unknowns, in addition to certain known data.
Solving an inequality consists of finding all the values of the unknown for which the inequality relation holds.
Inequality in this caseIn this case, you know that to fit in an existing frame, the length, x, of a piece of glass must be longer than 12 cm but not longer than 12.2 cm.
It means that the length has to be strictly greater than 12 cm but less than or equal to 12.2 cm. In other words, length of frame must be longer than 12 cm means that x>12, while ength of frame not longer than 12.2 cm means x≤12.2
Finally, the inequality that expresses the previous relationship is
12 < x ≤ 12.2
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Pmlease helppp tyyyy
Answer:
Step-by-step explanation:
Solving quadratic equation using formula:5x² - 6x - 2 = 0
a = Coefficient of x² = 5
b = coefficient of x = -6
c = constant = -2
Discriminant = D = b² - 4ac
= (-6)² - 4 * 5 * (-2)
= 36 + 40
= 76 > 0
So, the quadratic equation has two distinct real roots.
[tex]\sf \boxed{\bf x = \dfrac{-b \± \sqrt{D}}{2a}}\\\\[/tex]
[tex]\sf = \dfrac{-[-6] \± \sqrt{76}}{2*5}\\\\= \dfrac{6 \±8.7}{10}\\\\=\dfrac{6+8.7}{10} \ ; \ or \ x =\dfrac{6-8.7}{10}\\\\=\dfrac{14.7}{10} \ ; \ or \ x = \dfrac{-2.7}{10}[/tex]
x = 1.47 or x = -0.27
Answer: x = 1.5 or (-0.3)
b) x² + 3x = 40
x² + 3x - 40= 0
We can solve the quadratic equation using factor method.
Sum = 3
Product = -40
Factors = 8 , (-5)
When we add, 8 + (-5) = 3 , we get 5and when we multiply 8 *(-5) = -40, we get -(-40).
Rewrite the middle term of the quadratic equation using the factors.
x² + 8x - 5x - 40 = 0
x(x + 8) - 5(x + 8)= 0
(x + 8) (x - 5) = 0
x + 8 = 0 or x - 5 = 0
x = -8 or x = 5
Answer : x = -8 , 5
There are federal excise taxes on the retail price when purchasing fishing equipment. the taxes are intended to help pay for parks and conservation. what is the federal excise tax rate if there is an excise tax of $17.50 on a fishing rod and reel that has a retail price of $175?
Using proportions, it is found that the federal tax rate is of 10%.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
The tax rate is the proportion that $17.50 is of $175, hence:
r = 17.50/175 = 0.1 = 10%.
The federal tax rate is of 10%.
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4 kilograms of almonds cost $34. how much does it cost for kilograms of almonds?
The cost of kilograms of almonds is $8.5 if 4 kilograms of almonds cost $34.
According to the questions,
4 kilograms of almonds cost $34. In order to find one kilograms of almonds only if we divide 34 by 4.
= [tex]\frac{34}{4}[/tex] = 8.5
Hence, the cost of kilograms of almonds is $8.5 if 4 kilograms of almonds cost $34.
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In a recent baseball season, Ron was hit by pitches 19 times in 596 plate appearances during the regular season. Assume that the probability that Ron gets hit by a pitch is the same in the playoffs as it is during the regular season. In the first playoff series, Ron has 22 plate appearances. What is the probability that Ron will get hit by a pitch exactly once
The probability that Ron will get by a pitch exactly once is 0.351912.
Given that out of 596 plate appearance Ron was hit 19 times from pitches.
We have to find the probability that ron was hit by pitches in 22 plate appearances.
The value of probability lies between 0 and 1. The formula of calculating proabability is number of items/ total items.
We can find the required probability through binomial distribution which is
n[tex]C_{r}p^{r} (1-p)^{n-r}[/tex] where n is the number of trials and r is the successes.
Probability that Ron will be hit =19/596
=0.031
Probability that Ron will get hit by a pitch exactly once =P(X=1)
=[tex]22C_{1}(0.031)^{1} (1-0.031)^{21}[/tex]
=22!/1!21!*0.031*[tex](0.969)^{21}[/tex]
=22*0.031*0.516
=22*0.015996
=0.351912
Hence the probability that Ron will get hit by a pitch exactly once is 0.35.
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what is the value of x
find the value of f
Answer:
x = 20°
f = 40°
Step-by-step explanation:
4x + 5x = 180°
because a straight line is 180°
so 9x = 180°
1x = 180 ÷ 9
1x = 20
all angles in a triangle add together is 180°
so 7x + f = 180°
1x = 20
7x = 20 × 7
7x = 140°
f = 180° - 140 °
f = 40°
Answer:
x = 20 , f = 40
Step-by-step explanation:
4x and 5x are a linear pair and sum to 180° , then
4x + 5x = 180
9x = 180 ( divide both sides by 9 )
x = 20
then
5x = 5 × 20 = 100° and 3x = 3 × 20 = 60°
the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.
5x is an exterior angle of the triangle , then
f + 3x = 5x , that is
f + 60 = 100 ( subtract 60 from both sides )
f = 40
The average rainfall in inches in miami from january to may is 1.61, 2.24, 2.99 3.15 5.35 and 9.68 while it is 2.24 2.8 3.03 2.05 2.09 and 6.69 during the same period in tampa. which city gets more rain and how much more.
Miami gets 6.12 more inches of rainfall more than Tampa.
What is average rainfall?The annual amount of precipitation that we anticipate is known as normal or typical rainfall (in a given area).
It is calculated and set by estimating the average (mean) of precipitation that has been observed in a region over a long period of time ( at least 30 years). The total amount of daily precipitation in a year is referred to as annual precipitation.
Calculation for the average rainfall-
The average rainfall (in inches) in Miami for 5 months are-
1.61, 2.24, 2.99, 3.15, 5.35, and 9.68,
Thus, the total rainfall in Miami in 5 months,
Total average rainfall = 1.61 +2.24 +2.99 +3.15 +5.35 + 9.68 = 25.02 inches
Similarly,
The average rainfall (in inches) in Tampa for 5 months are-
2.24, 2.8, 3.03, 2.05, 2.09, and 6.69.
Thus, the total rainfall in Tampa in 5 months,
Total average rainfall = 2.24 +2.8 +3.03 + 2.05 + 2.09 + 6.69 = 18.9 inches.
The difference between the total rainfall in Miami and Tampa is
25.02 - 18.9 = 6.12 inches
The total rainfall in Miami is 25.02 inches while the total rainfall in Tampa in the same time period is 18.9 inches.
Therefore, Miami gets 6.12 inches more rainfall during the same time period.
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Instructions: Find the missing length indicated.
400
144
X
The missing side of a right triangle x is 24 units.
How to find side of a right angle triangle?A right triangle has one of its angles as 90 degrees. The side of a right angle triangle can be found using trigonometric ratios.
The side x can be found using similar triangle ratios,
Hence,
x / 400 = 144 / x
cross multiply
x × x = 400 × 144
x² = 57600
square root both sides
√x² = √57600
x = 240
Therefore, the value of x in the right triangle is 240 units.
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The sum of the reciprocal of a positive number and the reciprocal of 3 more than the number is
7/10. Find the number.
Write an equation with the following characteristics:
1. It uses x to represent the number.
2. It uses the information as it is given above.
3. It can be solved to answer the question.
Equation:
The number is:
Answer:
[tex]\sf Equation: \quad \dfrac{1}{x}+\dfrac{1}{x+3}=\dfrac{7}{10}[/tex]
The number is 2.
Step-by-step explanation:
The reciprocal of a number is simply 1 divided by the number.
The reciprocal of a fraction is where the numerator and denominator are swapped (so the fraction is turned upside down).
Let x be the positive number.
The reciprocal of the positive number x:
[tex]\sf \implies \dfrac{1}{x}[/tex]
The reciprocal of 3 more than the number x:
[tex]\sf \implies \dfrac{1}{x+3}[/tex]
The sum of the reciprocal of a positive number and the reciprocal of 3 more than the number is 7/10:
[tex]\implies \sf \dfrac{1}{x}+\dfrac{1}{x+3}=\dfrac{7}{10}[/tex]
To solve, make the denominators of the two fractions on the left side the same by multiplying them.
Multiply the numerator of the first fraction by the denominator of the second fraction to get the new numerator of the first fraction.
Multiply the numerator of the second fraction by the denominator of the first fraction to get the new numerator of the second fraction.
[tex]\implies \sf \dfrac{1(x+3)}{x(x+3)}+\dfrac{1(x)}{(x+3)(x)}=\dfrac{7}{10}[/tex]
[tex]\implies \sf \dfrac{x+3}{x(x+3)}+\dfrac{x}{x(x+3)}=\dfrac{7}{10}[/tex]
Add the numerators and keep the denominator so that there is now one fraction:
[tex]\implies \sf \dfrac{2x+3}{x(x+3)}=\dfrac{7}{10}[/tex]
Cross multiply:
[tex]\implies \sf 10(2x+3)=7x(x+3)[/tex]
Expand:
[tex]\implies \sf 20x+30=7x^2+21x[/tex]
Rearrange so the equation is equal to zero:
[tex]\implies \sf 7x^2+x-30=0[/tex]
Split the middle term:
[tex]\implies \sf 7x^2+15x-14x-30=0[/tex]
Factor the first two terms and last two terms separately:
[tex]\implies \sf x(7x+15)-2(7x+15)=0[/tex]
Factor out the common term (7x + 15):
[tex]\implies \sf (x-2)(7x+15)=0[/tex]
Apply the zero product property:
[tex]\sf (x-2)=0 \implies x=2[/tex]
[tex]\sf (7x+15)=0 \implies x=-\dfrac{15}{7}[/tex]
As x is a positive number, x = 2.
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For 3, 5, Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer.
8. What do you know about triangles PDX and triangle CRX? Justify your answer.
Answer:
The triangles are Congruent for the same reason as the answer above
Step-by-step explanation:
The slope in the simplest form
La suma de un número con su mitad es igual a 75. ¿Cuál es ese número?
Doy corona porfis
Considerando la definición de una ecuación, el número desconocido es 50.
Definición de ecuaciónUna ecuación es la igualdad existente entre dos expresiones algebraicas conectadas a través del signo de igualdad en la que figuran uno o varios valores desconocidos, llamadas incógnitas, además de ciertos datos conocidos.
Los miembros de una ecuación son cada una de las expresiones que aparecen a ambos lados del signo igual.
Número desconocidoEn este caso, "La suma de un número con su mitad es igual a 75" puede ser expresado mediante una ecuación, donde consideras que "x" es la incógnita, que representa el número desconocido.
Teniendo en cuenta que la mitad de un número se calcula dividiendo el número por 2, entonces la ecuación queda expresada como:
x + x÷2= 75
O lo que es lo mismo
x + [tex]\frac{1}{2}[/tex]x= 75
Resolviendo:
[tex]\frac{3}{2}[/tex]x= 75
x= 75 ÷[tex]\frac{3}{2}[/tex]
x= 75×[tex]\frac{2}{3}[/tex]
x=50
Finalmente, el número desconocido es 50.
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From Andys house to Billys hometown you can travel by 3 roads. And to get from Billys hometown to Willies's house you can travel by 5 roads. How many possible wats are there to travel from andys house to willies house?
3 roada andy to billy
5 road billy to willies
3×5 =15 it's the answer 15
Total area=
Help me please!! Thanks so much