Given the disk of the radius r = 1, i.e., = {(x₁, x₂) € R² | x² + x² <1} find the smallest and largest values that the function f(x₁, x₂) = x₁ + x₂ achieves on the set D. a) Formulate the problem as an optimization problem and write down the optimality conditions. b) Find the point(s) in which the function f achieves maximum and minimum on the set D What is the largest and smallest value of f ? Comments: Make sure that you properly justify that you find a minimizer and maximizer. c) Denote the smallest value fin. What is the relative change of fin expressed in percents if the radius of the disk decreases and it is given as D {(1,₂) € R²|x²+x≤0.99}
Answers
The smallest value of the function f(x₁, x₂) = x₁ + x₂ on the disk D with a radius of 1 is -√2, and the largest value is √2. The relative change in the smallest value, expressed in percent, can be calculated if the radius of the disk decreases to 0.99.
a) The problem can be formulated as an optimization problem with constraints. We want to find the smallest and largest values that the function f(x₁, x₂) = x₁ + x₂ achieves on the set D, which is defined as the disk with radius r = 1, i.e., D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² < 1}.
To find the smallest value, we can minimize the function f subject to the constraint that (x₁, x₂) is within the disk D. Mathematically, this can be written as:
Minimize: f(x₁, x₂) = x₁ + x₂
Subject to: x₁² + x₂² < 1
To find the largest value, we can maximize the function f subject to the same constraint. Mathematically, this can be written as:
Maximize: f(x₁, x₂) = x₁ + x₂
Subject to: x₁² + x₂² < 1
b) To find the points at which the function f achieves the maximum and minimum on the set D, we can analyze the problem. The function f(x₁, x₂) = x₁ + x₂ represents a plane with a slope of 1.
Considering the constraint x₁² + x₂² < 1, we observe that it represents a circle with radius 1 centered at the origin.
Since the function f represents a plane with a slope of 1, the maximum and minimum values occur at the points on the boundary of the disk D where the plane is tangent to the disk. In other words, the maximum and minimum values occur at the points where the plane f(x₁, x₂) = x₁ + x₂ is perpendicular to the boundary of the disk.
Considering the disk D: x₁² + x₂² < 1, we can see that the boundary of the disk is x₁² + x₂² = 1 (the equation of a circle).
At the boundary, the gradient of the function f(x₁, x₂) = x₁ + x₂ is parallel to the normal vector of the boundary circle. The gradient of f is (∂f/∂x₁, ∂f/∂x₂) = (1, 1), which represents the direction of steepest ascent of the function.
Thus, at the points where the plane f(x₁, x₂) = x₁ + x₂ is tangent to the boundary circle, the gradient of f is parallel to the normal vector of the circle. Therefore, the gradient of f at these points is proportional to the vector pointing from the origin to the tangent point.
To find the tangent points, we can use the fact that the tangent line to a circle is perpendicular to the radius at the point of tangency. The radius of the circle D is the vector from the origin to any point (x₁, x₂) on the boundary, which is (x₁, x₂).
So, the tangent points occur when the gradient vector (1, 1) is proportional to the radius vector (x₁, x₂), which means:
1/1 = x₁/1 = x₂/1
Simplifying, we get:
x₁ = x₂
Substituting this back into the equation of the boundary circle, we have:
x₁² + x₂² = 1
x₁² + x₁² = 1
2x₁² = 1
x₁² = 1/2
Taking the positive square root, we get:
x₁ = √(1/2)
Since x₁ = x₂, the corresponding values are:
x₂ = √(1/2)
Thus, the points where the function f achieves the maximum and minimum on the set D are (x₁, x₂) = (√(1/2), √(1/2)) and (x₁, x₂) = (-√(1/2), -√(1/2)).
Plugging these values into the function f(x₁, x₂) = x₁ + x₂, we get:
f(√(1/2), √(1/2)) = √(1/2) + √(1/2) = 2√(1/2) = √2
f(-√(1/2), -√(1/2)) = -√(1/2) - √(1/2) = -2√(1/2) = -√2
Therefore, the largest value of f is √2, and the smallest value of f is -√2.
c) Denoting the smallest value as fin = -√2, we can find the relative change in fin expressed in percent if the radius of the disk decreases to D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² ≤ 0.99}.
To calculate the relative change, we can use the formula:
Relative Change = (New Value - Old Value) / Old Value * 100
The new value of fin, denoted as fin', can be found by minimizing the function f subject to the constraint x₁² + x₂² ≤ 0.99.
Solving the minimization problem, we find the new smallest value fin' on the set D with a radius of 0.99.
Comparing fin' to fin, we can calculate the relative change:
Relative Change = (fin' - fin) / fin * 100
By solving the new minimization problem, you can find the new smallest value fin' and calculate the relative change using the formula provided.
To learn more about functions visit : https://brainly.com/question/11624077
#SPJ11
Related Questions
Solve each system by substitution
2x+y=20
6x-5y=12
what is (x,y)?
Answers
Answer:
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point Form:(7,6)
Equation Form: x=7, y=6
2x+y=20
6x-5y=12
First get y by its self
2x+y=20
y= -2x+20
Now plug into second equation for y
6x-5(-2x+20)=12
6x+10x-100
16x-100=12
16x=112
x=7
now plug in x value into first equation and solve for y
2(7)+y=20
14+y=20
y= 6
Now check by plugging x and y values back into equation
2(7)+6=20
6(7)-5(6)=12
They both work so x=7 and y =6
x/7 ≥ − 6=============
Answers
The range of value of x in the inequality x/7 ≥ -6 is x≥ -42
What is inequality?Inequality, is a statement of an order relationship. Some terms used in inequality are ,greater than, greater than or equal to, less than, or less than or equal to. They are used between two numbers or algebraic expressions.
greater than has the sign >
greater than or equal to has the sign ≥
less than has the sign < and
less than or equal to has ≤
solving the range of value of x in the inequality x/7 ≥- 6
multiply both sides by 7
x ≥ -42
Therefore the range of value of x is x ≥ -42
learn more about inequality from
https://brainly.com/question/24372553
#SPJ1
a binomial experiment with probability of success =p0.7 and =n7 trials is conducted. what is the probability that the experiment results in exactly 6 successes?
Answers
The probability that the experiment results in exactly 6 successes is 0.2668.
To calculate the probability of exactly 6 successes in a binomial experiment with p=0.7 and n=7, we can use the binomial probability formula:
P(X = k) = (n choose k) * \(p^{k}\) * \((1-p)^{n-k}\)
where X is the random variable representing the number of successes, k is the number of successes we're interested in, and n is the total number of trials.
Substituting the values, we get:
P(X = 6) = (7 choose 6) * \(0.7^{6}\) * \((1-0.7)^{7-6}\)
= 7 * \(0.7^{6}\) * \(0.3^{1}\)
= 0.266827932
Therefore, the probability of exactly 6 successes in a binomial experiment with p=0.7 and n=7 is approximately 0.2668.
To learn more about probability here:
https://brainly.com/question/32004014
#SPJ4
angles L and M are supplementary what are the some of their measures?
Answers
Answer:
Supplementary angles add upto 180 degrees. Since angles L and M are supplementary, the sum the two angles must be 180 degrees.
Step-by-step explanation:
2m + 3m2 - 4m
What is if if I simplify each expression by combining like terms
Answers
Answer:
= 3m2 + 2m - 4m
= 3m2 - 2m
Step-by-step explanation:
you can only subtract similar products
will give u brainliest!
plz help me out
Answers
Answer:
The third one i think
Step-by-step explanation:
Answer:
its the 4th one, the onlyone that makes sense
Step-by-step explanation:
the 3rd one cant since a function always had one input for every output
PLEASE HELP ASAP!
Which expression is equivalent to 5^10 x 5^5?
5^2
5^5
5^15
5^50
Answers
Answer:
5^15
Step-by-step explanation:
5^5 is 3125, and 5^10 is 9765625. When multiplied, you get 30517578125. 5^5 is in the equation, so that couldn't be the answer. 5^50 is too big of a number, so that also wouldn't work. 5^2 is 25, and that would be too small. That leaves 5^15, which when calculated leads to 30517578125.
Answer:
5^15
Step-by-step explanation:
5^10 is equivalent to 9765625 and 5^5 is equivalent to 3125. 9765625x3125 is equal to 30517578125. We need to find the expression wich is equal to that number. It can't be 5^2 (25), 5^5 (3125), 5^50 (8.8817842e+34). The answer is 5^15 (30517578125)
Simplify the following expression using exponents
Answers
Answer:
-16 over 125a^11b
Step-by-step explanation:
p = 2i+ 2w
Find p if i = 79 and w = 21.
p=
Answers
Answer:
200
Step-by-step explanation:
\(p=2\cdot \:79+2\cdot \:21\\2\cdot \:79=158\\p=158+2\cdot \:21\\2\cdot \:21=42\\p=158+42\\p=200\)
Which segment is an angle bisector of triangle ABC?
A. BX
B. SR
C. BF
D. AR
Answers
Answer: A. BX
Step-by-step explanation:
BX is the only line that bisects an angle. BX bisects angle B. Therefore, BX is an angle bisector.
SR is an altitude
BF is a segment of BC
AR is a chord
BX is the angle bisector of triangle ABC.
Angle bisecting refers to dividing an angle into two equal or congruent parts. The line or ray that divides the angle into two equal parts is called the angle bisector. It always cuts the angle into two angles of equal measure.
The properties of an angle bisector are :
It divides the angle into two equal angles, i.e., both resulting angles have the same measurement.
It is equidistant from the two sides of the angle, i.e., any point on the angle bisector is equidistant from the sides of the angle.
If a point lies on the angle bisector, it is equidistant from the two sides of the angle.
So, looking at the properties we can say that BX is the angle bisector of triangle ABC.
Learn more about angle bisector
https://brainly.com/question/28565813
#SPJ2
Gabriel kicks a football. Its height in feet is given by h(t) = -16t² + 88t where t
represents the time in seconds after kick. What is the appropriate domain for this
situation?
Answers
The domain of the function h(t) = -16t² + 88t is equal to [0 , 5 ].
Function is equal to,
h(t) = -16t² + 88t
Where 't' represents the time in seconds after kick
The domain of a function is the set of all possible values of the independent variable for which the function is defined.
Only independent variable is t.
And there are no restrictions on its value.
Since the function represents the height of a football in feet.
The domain should be restricted to the time when the ball is in the air.
From the time of the kick until the time when the ball hits the ground.
The ball hits the ground when its height is 0.
So, the function h(t) = 0
Solve for t to get the time when the ball hits the ground,
⇒ -16t² + 88t = 0
⇒ -16t(t - 5.5) = 0
⇒ t = 0 or t = 5.5
The ball is kicked at t = 0.
So the appropriate domain for this situation is,
0 ≤ t ≤ 5.5
Therefore, the appropriate domain of the function h(t) is for all values of t between 0 and 5.5 seconds (inclusive).
learn more about domain here
/brainly.com/question/30944862
#SPJ4
What table of values goes with the equation y = |-2x|?
Answers
Answer:
The graph at the bottom
Step-by-step explanation:
All the numbers are gonna come out positive, because of the absolute value bars.
Answer:
i think its the last one i got that answer right
Step-by-step explanation:
Determine the least three-digit number that is divisible by 3,5 and 9
Answers
135
because 5 • 3 • 9 is 135 and there isn't a three digit number divisible by each of the three numbers
Which pair of triangles can be proven congruent by SAS?
Answers
Answer:
i don't know
Step-by-step explanation:
i am lower grade srry
y=
(x^2)/(x^3-4x)
please provide mathematical work to support solutions.
e) Find the first derivative. f) Determine the intervals of increasing and decreasing and state any local extrema. g) Find the second derivative. h) Determine the intervals of concavity and state any
Answers
The first derivative is e) Y' = [-x⁴ - 4x²] / (x³ - 4x)².
f) The function Y = (x²) / (x³ - 4x) is increasing on the intervals (-∞, 0) and (2, ∞) and decreasing on the interval (0, 2); it does not have any local extrema.
g) The second derivative of Y = (x²) / (x³ - 4x) is Y'' = [-4x³ - 8x](x³ - 4x)² + (-x⁴ - 4x²)(3x² - 4)(x³ - 4x) / (x³ - 4x)⁴.
h) The intervals of concavity and any inflection points for the function Y = (x²) / (x³ - 4x) cannot be determined analytically and may require further simplification or numerical methods.
How to find the first derivative?
e) To find the first derivative, we use the quotient rule. Let's denote the function as Y = f(x) / g(x), where f(x) = x² and g(x) = x³ - 4x. The quotient rule states that (f/g)' = (f'g - fg') / g². Applying this rule, we have:
Y' = [(2x)(x³ - 4x) - (x²)(3x² - 4)] / (x³ - 4x)²
Simplifying the expression, we get:
Y' = [2x⁴ - 8x² - 3x⁴ + 4x²] / (x³ - 4x)²
= [-x⁴ - 4x²] / (x³ - 4x)²
f) To determine the intervals of increasing and decreasing and identify any local extrema, we examine the sign of the first derivative. The numerator of Y' is -x⁴ - 4x², which can be factored as -x²(x² + 4).
For Y' to be positive (indicating increasing), either both factors must be negative or both factors must be positive. When x < 0, both factors are positive. When 0 < x < 2, x² is positive, but x² + 4 is larger and positive. When x > 2, both factors are negative. Therefore, Y' is positive on the intervals (-∞, 0) and (2, ∞), indicating Y is increasing on those intervals.
For Y' to be negative (indicating decreasing), one factor must be positive and the other must be negative. On the interval (0, 2), x² is positive, but x² + 4 is larger and positive.
Therefore, Y' is negative on the interval (0, 2), indicating Y is decreasing on that interval.
There are no local extrema since the function does not have any points where the derivative equals zero.
g) To find the second derivative, we differentiate Y' with respect to x. Using the quotient rule again, we have:
Y'' = [(d/dx)(-x⁴ - 4x²)](x³ - 4x)² - (-x⁴ - 4x²)(d/dx)(x³ - 4x)² / (x³ - 4x)⁴
Simplifying the expression, we get:
Y'' = [-4x³ - 8x](x³ - 4x)² + (-x⁴ - 4x²)(3x² - 4)(x³ - 4x) / (x³ - 4x)⁴
h) To determine the intervals of concavity, we examine the sign of the second derivative, Y''. However, the expression for Y'' is quite complicated and difficult to analyze analytically.
It might be helpful to simplify and factorize the expression further or use numerical methods to identify the intervals of concavity and any inflection points.
Learn more about first derivative on:
https://brainly.com/question/10023409
#SPJ4
What is the solution to this equation? 4- 5k = -21
a.5
b.21
c.17 over 5
4 over 5
Answers
Answer:
k = 5
Step-by-step explanation:
4 - 5k = -21
Transpose and subtract 4 from both sides:
4 - 4 - 5k = -21 -4
-5k = -25
Divide both sides by -5
-5k/-5 = -25/-5
k = 5
if x = 7 and f = 4 and g = 9 and l = 24 and p = 12 and e = 9847493 then what is x ÷ fg X i ÷ p
Answers
the election of 2000 demonstrated that a poll isn't reliable if which of the following occurs?responsesthe election is too close to call.the election is too close to call.the sample is biased.the sample is biased.the sample is too small.the sample is too small.too many states are red.
Answers
The election of 2000 demonstrated that a poll may not be reliable if the sample size is too small (option d).
The factor that can impact the reliability of a poll is bias. If the sample is biased, it may not accurately represent the larger population, leading to skewed results. Bias can occur in several ways, such as selecting a sample that is not representative of the larger population, asking leading questions, or using a sampling method that favors a particular group.
In the 2000 election, both of these factors contributed to the unreliability of the polls. The race was extremely close, with the outcome depending on the results in a few key states. Pollsters struggled to accurately predict the outcome of the election, with some predicting a win for Al Gore and others predicting a win for George W. Bush.
Additionally, the sample sizes and methods used by pollsters were called into question. Some pollsters used small sample sizes, while others were accused of bias in their sampling methods. The combination of these factors led to unreliable poll results and uncertainty about the outcome of the election.
Hence the correct option is (d).
To know more about sample here
https://brainly.com/question/13287171
#SPJ4
what's the value of :-
\(( \sqrt[ 3]{343} ){}^{2} \)
Answers
Answer:
49
hope it helps, kaul
thanks
Answer:
49
Step-by-step explanation:
\( { (\sqrt[3]{343} )}^{2} \)
(B.E.D.M.A.S rule) [First within the brackets]
\({ (\sqrt[3]{7 \times 7 \times 7} )}^{2} \)
\( {7}^{2} \)
[Then exponents]
= 49 (Ans)
A research group developed the following mathematical model relating systolic blood pressure and age: P(x)=a+ bln(x+1). Where P(x) is pressure, measured in millimeters of mercury, and x is age in years. By examining Guilford County hospital records, they estimate the values for Guilford County to be a=43 and b=25. Using this model, estimate the rate oI change of pressure with respect to time after 31 years. Round to the nearest hundredth ( 2 decimal places). ____ millimeters per year
Answers
Rounding to two decimal places, the rate of change of blood pressure with respect to time after 31 years is approximately 0.81 millimeters per year.
The mathematical model relating systolic blood pressure and age is given as P(x)=a+b*ln(x+1), we can differentiate it with respect to time (t) to find the rate of change of pressure with respect to time:
dP/dt = dP/dx * dx/dt
Here dx/dt is the rate of change of age with respect to time, which is simply 1 year per year or 1.
Taking the derivative of P(x) with respect to x, we get:
dP/dx = b/(x+1)
Substituting the given values for a and b, we have:
P(x) = 43 + 25ln(x+1)
dP/dx = 25/(x+1)
Therefore, the rate of change of pressure with respect to time after 31 years is:
dP/dt = dP/dx * dx/dt = (25/(31+1)) * 1 = 0.8065
Learn more about mathematical model visit: brainly.com/question/28592940
#SPJ4
HELP ASAP I DONT WANT TO FAIL
Answers
Answer:
16.3p + (-10)
Step-by-step explanation:
Please help I will give brainiest
Answers
Answer: it shows the answer i think
Step-by-step explanation:
On the side where there is F for farenight where the red mark stops it says 50 for problem one, so that is the answer for Farenheight. For celecius, 10 is the answer, because the point stops there. If you dont understand here are answers. (i think are correct im not sure but they should be)
4. F = 50 C=10 5. C= 20 F= 69 6. C = 5 F= 40
Please help find the equation and tell me how you found it! If you don't show work I swear I'll report it and don't you dare give the wrong answer. i ain't giving 40 points for no reason.
Answers
Answer:
y=-\(\sqrt9(x-1)+2\)
The quotient of 1/2 divided by 1 3/4
Answers
Answer:
2/7
Step-by-step explanation:
1
2
÷
1 3
4
=
1
2
÷
7
4
=
1
2
×
4
7
=
1 × 4
2 × 7
=
4
14
=
4 ÷ 2
14 ÷ 2
=
2
7
Which is a way to find the equation of a line that best fits this data?
Answers
Whereas the data is not provided, keep in mind that the concept necessary to identify the Line of Best Fit is the same across all statistical and mathematical issues where data must be visually analyzed for correlation. This is how.
How do you determine the optimal fit line?The most basic method for determining the line of greatest fit is to use a regressive analysis calculator. Knowing what the computer is doing, on the other hand, necessitates understanding the reasoning behind the process. Anyone with a piece of paper and a pencil can compute it once they grasp the technique.
Begin by plotting the collected data on a scatter graph. This is crucial since it sets and arranges the formula's needed values. The following formula is used to compute the line of best fit:
Y = C +B¹(x¹) + B²(x²)
In this example, Y is the equation's dependent variable.C is unchanging or constant.B¹ and B² are the first and second regression coefficients, respectively.The first and second independent variables are X¹ and X²After the points have been plotted, a line that maintains a balance of the points on both sides, as shown in the attached example, must be created to demonstrate the nature of the data. It is linear in the case of the attached example.
Learn more about line of best fit;
https://brainly.com/question/19355376
#SPJ1
A spinner has spaces numbered 1 through 10, with an equal chance of
landing on each. A winning spin is one that lands on a 7 or more. What is
the probability of a winning spin? (answer as a reduced fraction) *
1/4
1/2
3/10
2/5
Answers
2/5 is the aneswer. beacuse 7/8/9/10 eqaul 4 times out of 10 to win 4/10 reduces down to 2/5.
let e be a lamina that lies below the cone between the spheres and in the first octant
Answers
The statement "Let e be a lamina that lies below the cone between the spheres and in the first octant" describes a specific region in three-dimensional space.
The region, denoted as e, is located below the cone formed by two intersecting spheres and is confined to the first octant. This means that e exists within the space where all coordinates are positive (x > 0, y > 0, z > 0). Further details about the specific dimensions, shape, or properties of the lamina are not provided in the given statement.
A lamina is a term used in mathematics to refer to a two-dimensional object, typically with a uniform density, that lies within a specific region of three-dimensional space. In this case, the lamina denoted as e is positioned below the cone formed by the intersection of two spheres. This indicates that the lamina occupies the space between the two spheres and extends downwards. Additionally, the lamina is confined to the first octant, which means that it is restricted to the region where all coordinates (x, y, and z) are positive.
However, without further information regarding the dimensions, shape, or specific properties of the lamina, it is not possible to provide a more detailed explanation. The given statement simply defines the region in three-dimensional space where the lamina, denoted as e, is located.
To learn more about octant: -brainly.com/question/30888654
#SPJ11
Answer the following.(a) A certain solution has a hydrogen ion concentration of 0.00000896 moles per liter. Write this number in scientific notation.(b) A humpback whale can weigh up to 1.1 * 10^5 pounds. Write this number in standard notation.
Answers
Given:
Two statements (a) and (b) are given.
Find:
we have to write the answers of both statments (a) and (b).
Explanation:
(a) A certain solution has a hydrogen ion concentration of 0.00000896 moles per liter
The scientific notation of 0.00000896 is
\(8.96×10^{-6}\)......................................................................................................................................
(b) A humpback whale can weigh up to 1.1 * 10^5 pounds
The standard form of the given number is given as below
\(1.1\times10^5=110000\)
hii please help i’ll give brainliest!!
Answers
Answer:
the answer is c
Step-by-step explanation:
plato users
How many centimeters are equal to 12 meters?
Answers
Hi!1200 centimetres are equal to 12 meters.
Answer:
1200cm
Step-by-step explanation:
1m = 100cm
12m = 1200cm