We first compute the partial derivatives of f(x, y):
f(x, y) = 5x² + 5y² - 10x + 21
∂f/∂x = 10x - 10
∂f/∂y = 10y
The critical points of f occur where both ∂f/∂x and ∂f/∂y are equal to zero. This only happens at the point (x, y) = (1, 0). (And this point does lie inside R.)
Next, compute the Hessian matrix H(x, y) for f :
[tex]H(x, y) = \begin{bmatrix}\frac{\partial^2f}{\partial x^2} & \frac{\partial^2f}{\partial x\partial y} \\ \frac{\partial^2 f}{\partial y\partial x} & \frac{\partial^2f}{\partial y^2}\end{bmatrix} = \begin{bmatrix}10&0\\0&10\end{bmatrix}[/tex]
Since det(H(x,y)) = 100 is positive for all x and y, this means the critical point (1, 0) is a local minimum, and we have f(1, 0) = 16.
Next, we check for extrema along the boundary of R, which is comprised of a semicircle with radius 2 and the line segment connecting (-2, 0) and (2, 0).
• Parameterize the semicircular portion by
x(t) = 2 cos(t)
y(t) = 2 sin(t)
with 0 ≤ t ≤ π. Then
f(x(t), y(t)) = 20 sin²(t) + 20 cos²(t) - 20 cos(t) + 21
which is simplifies to a function of one variable t,
g(t) = 41 - 20 cos(t)
Find the extrema of g over the interval [0, π] : we have critical points when
g'(t) = 20 sin(t) = 0
which happens at t = 0 and t = π. At these points, we get local extreme values of
•• t = 0 => x = 2 and y = 0 => f(2, 0) = 21
•• t = π => x = -2 and y = 0 => f(-2, 0) = 61
• Over the line-segment portion, we take y = 0, so f(x, y) again reduces to a function of one variable:
f(x, 0) = 5x² - 10x + 21
Completing the square, we have
5x² - 10x + 21 = 5 (x - 1)² + 16
which has a maximum value of 16 when x = 1 (and this happens to be the critical point (1, 0) we found earlier).
So, over the region R, f(x, y) has
• an absolute maximum of 61 at the point (-2, 0), and
• an absolute minimum of 16 at (1, 0)
Robert has 4 times as many pennies as he does nickels. If Robert has $1.44 worth of pennies and nickels, how many nickels does he have?
Can some please help me I will mark u brilliant
Question 2: -4/3
Question 3: -7/20
Five students share the writing of a term
paper evenly. If each person writes more
than six pages, how long is the paper?
Answer:
Over 30
Step-by-step explanation:
There is no definite answer if you don't know how many pages each person. All you have to do is multiply 5 by 6
the probability that a tin of paint is dented is 0.07 out of 3000 tins of paint,how many would you expect to be dented?
Answer:
3000×0.07=210 tins
Step-by-step explanation:
210 tins
Step-by-step explanation:
3000÷100×0.07
=2.1
MARK ME BRAINLIEST PLS!!What is 11 divided by 1/4
Find one value of x for F(x)=-4
F(x) = -4 means y = -4. That means whatever the x is, the answer is going to be -4.
which of the following does not require the use of chain rule to find dy/dx.
C. For each term in this expression, you can just apply the power rule.
[tex]y = 3x^2 - \sqrt x + \dfrac2x = 3x^2 - x^{\frac12} + 2x^{-1}[/tex]
[tex]\implies \dfrac{dy}{dx} = 2\cdot3x^{2-1} - \dfrac12 x^{\frac12-1} + (-1)\cdot2x^{-1-1}[/tex]
[tex]\implies \dfrac{dy}{dx} = 6x - \dfrac12 x^{-\frac12} - 2x^{-2}[/tex]
[tex]\implies \dfrac{dy}{dx} = 6x - \dfrac1{2\sqrt x} - \dfrac2{x^2}[/tex]
Every other choice involves composite functions that do require the chain rule to differentiate.
Option C : [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex] can be differentiated without using chain rule.
The differentiation of [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex] without using chain rule can be done as follows:
[tex]\begin{aligned}\dfrac{dy}{dx} &= \dfrac{d}{dx}(3x^2 - \sqrt{x} + \dfrac{2}{x})\\\\&= \dfrac{d}{dx}(3x^2 - x^{\frac{1}{2}} + 2x^{-1})\\&= 6x - (1/2)x^{-\frac{3}{2}} -2x^{-2}\\\end{aligned}[/tex]
The rest of the functions are composite, which means, that after differentiating them with some other base, you need to change the base and differentiate again, thus applying chain rule.
Thus, the function [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex] can be differentiated without using chain rule.
Thus, option C: [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex] is correct option.
Learn more here:
https://brainly.com/question/5543508
What is the volume, in cubic inches, of a rectangular prism with a height of 7 inches, a width of 10 inches, and a length of 9 inches?
Answer:
Step-by-step explanation:
87
Answer:
630 in.³
Step-by-step explanation:
To find the volume of a rectangular prism, you can multiply the height, times the width, times the length. So, let's go ahead and do that with the values of the given prism.
7 × 10 × 9 = 70 × 9 = 630
So, our volume is 630...but not so fast. We also have to put in our units. Our units are inches × inches × inches, or inches³. So our final value is 630 in.³
Hopefully that's helpful! :)
HELP! Can anyone help me i need help it this question
Average rate = distance / time
Rewrite distance as an improper fraction":3 1/3 = 10/3
Now you have 10/3 / 1/3
When dividing by a fraction flip the second fraction over and multiply:
10/3 x 3/1 = (10 x 3) / (3/1) = 30/3 = 10
Average rate was 10 miles per hour
10% of $70 = what is the answer i do not now it.
Answer:
10% 0f 70 is 7
Step-by-step explanation:
70÷10=7
10%=7
Multiplying polynomials
Steps are in the attachment.
________
Hope it helps :)) ⚜
A company that bakes chocolate chip cookies averages 5.2 chocolate chips per cookie. Assume that the number of chocolate chips per cookie follows the Poisson distribution. What is the probability that a randomly selected cookie will contain two or more chocolate chips
Answer 12
Step-by-step explanation:
A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four units, and because each has identical packaging, the selection will be random. What is the probability that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good?
I hope that helped?
I need to know how to subtract fractions 4 2/3 - 3/4
Answer: The answer would be 3.91666666667
[tex]\huge\textsf{Hey there!}[/tex]
[tex]\mathsf{4 \dfrac{2}{3} - \dfrac{3}{4}}[/tex]
[tex]\mathsf{= \dfrac{4\times3+2}{3} - \dfrac{3}{4}}[/tex]
[tex]\mathsf{= \dfrac{12 + 2}{3} - \dfrac{3}{4}}[/tex]
[tex]\mathsf{= \dfrac{14}{3} - \dfrac{3}{4}}[/tex]
[tex]\mathsf{= \dfrac{56}{11} - \dfrac{9}{12}}[/tex]
[tex]\mathsf{= \dfrac{14\times4}{12 - 0}}[/tex]
[tex]\mathsf{= \dfrac{56 - 9}{12}}[/tex]
[tex]\mathsf{= \dfrac{47}{12}}[/tex]
[tex]\mathsf{= 3 \dfrac{11}{12}}[/tex]
[tex]\huge\textsf{Therefore, your answer should be: }\huge\boxed{\\\mathsf{\dfrac{47}{12} \ or\ 3 \dfrac{11}{12}}}\huge\checkmark[/tex]
[tex]\huge\textsf{Good luck on your assignment \& enjoy your day!}[/tex]
In grade 10, the ratio of boys to girls is 2: 3. There are 75 girls. How many are boys?
HELP PLEASE .. I WILL GIVE A BRAINLIEST !
Answer:
150
Step-by-step explanation:
The ration is 2/3 in this example means they are twice as many boys than girls. If boys = 2/3, than girls = 1/3.
So, to find the answer, you must multiply 75 by 2, which is 150.
You don't need this, but they are 75 + 150 = 225 boys and grils in grade 10.
The width of a rectangle is 6 kilometers
a
less than twice its length. If its area is
56 square kilometers, find the
dimensions of the rectangle.
Answer:
9,333333333333333 and 6 Km
Step-by-step explanation:
The width of a rectangle is 6 kilometers
less than twice its length. If its area is
56 square kilometers, find the
dimensions of the rectangle.
the area of the rectangle is found by multiplying the width by the height, you have the area and the width, divide the area by the width and find the length
56 : 6 = 9,333333333333333 Km
-----------------------------
check
9,333333333333333 * 6 = 56
the answer is good
Solve for x: -2(x + 3) = -2x - 6
A. 0
B. 3
C. All real numbers
D.No solution
Answer:
D.No solution
Answer:
C
Step-by-step explanation:
-2(x + 3) = -2x - 6 Remove the brackets
-2x - 6 = -2x - 6
Since the left and right sides are equal, any number will work.
If you wind up with a false statement, then there are no solutions.
2x + 4 = 2x + 7 Subtract 2x from both sides.
4 = 7
Find the length of the hypotenuse, side c, of a right triangle with legs of length a = 4 and b = 5. Use a calculator to estimate the
square root to one decimal place.
answer:
6.4
Step-by-step explanation:
4^2+5^2=c^2
16+25
square root of 41
c^2=6.4
Four subtracted from x is less than 18.
Answer:
x - 4 < 18
Step-by-step explanation:
< means less than
> means more than
here is the other question
Answer:
1. -3 × A1 + A2 --> B2
2. [tex]-\frac{1}{7}[/tex] × B2 --> C2
Step-by-step explanation:
1. -3 × A1 + A2 --> B2
(4x × -3) + 12x = -12x + 5x = -7x
(2y × -3) + 6y = -6y + 6y = 0y
(-10 × -3) + -9 = 30 - 9 = 21
2. [tex]-\frac{1}{7}[/tex] × B2 --> C2
-7x × [tex]-\frac{1}{7}[/tex] = 1x
21 × [tex]-\frac{1}{7}[/tex] = -3
Same process as last time
pls help with khan academy
Answer:
c
Step-by-step explanation:
because the rest are just way off
Answer:
The answer is C, bro
Step-by-step explanation:
hope this helped.
Umm how to do rates
The definition of a rate in math is, the speed of something being gained in a problem, let's say we have a hot pipe that can full up a pool in 6 hours, our rate would be 1/6 of the pool per hour.
in which system are the lines parallel ?
The graph of the function f(x) = (x + 2)(x + 6) is shown
below.
Which statement about the function is true?
O The function is positive for all real values of x where
x>-4.
6
14-
O The function is negative for all real values of x where
-6 < x < -2.
The function is positive for all real values of x where
x <-6 orx > -3.
O The function is negative for all real values of x where
x < -2.
24
-2
2
4
6
х
4
Answer:
The function is negative for all real values of x where
-6 < x < -2.
Step-by-step explanation:
f(x) = (x +2)(x + 6)
x + 2 = 0 or x + 6 = 0
x = -2 or x = -6
After graphing the function is negative between -6 < x < -2
The true statement about the graph is (a) The function is negative for all real values of x where -6 < x < -2.
The equation of the function is given as:
[tex]f(x) = (x + 6)(x + 2)[/tex]
From the graph, we have the following highlights
The values of the function decreases from negative infinity till x = -4The values of the function increases from x = -4The values of the function is negative for x = -6 to x = -2The above means that:
The function is negative for all real values of x where -6 < x < -2.
Hence, the true statement about the graph is (a)
Read more about quadratic graphs at:
https://brainly.com/question/7988424
We are flying at 30 000 feet.
Our speed is 500 miles per hour.
Our total flight time will be 10 hours and 20 minutes.
Mark knows the distance from London to Cancun is 4944 miles. He asks his daughter to check the information given by the pilot. She knows this formula for speed.
s = dt
s = speed (miles per hour) d = distance (miles)
t = time (hours)
Evaluate the information given by the pilot.
Show calculations to support your answer.
Answer:
Step-by-step explanation:
500 mph with a flight time of 10 hours and 20 minutes [(20 min)*(1 h/60 min) = 0.333 hr]
Distance = Speed * Time
Time = 10.333 hours
Distance = (500 miles/hr)*(10.333 hrs) = 5166.5 miles
The pilot's numbers indicate a distance of 5167 miles, compared to Mark's number of 4944 miles. That is a 4.3% reduction from the pilot's numbers. Not too bad, so we might question 1) Mark's knowledge, or 2) the impact that the approach and landing speeds have of the total miles flow.
My preference would be to trust the pilot's data. I don't know where Mark got his math education.
what is 1 1/2 of 150 and what is 1/2 of 180
Answer:
1 1/2 of 150 is 225 and 1/2 of 180 is 90
Step-by-step explanation:
1 1/2 of 150 basically means 1 1/2 x 150, which equals to 225. 1/2 of 180 means 1/2 x 180 which is 90.
Answer:
1 1/2 of 150 is "225" and
1/2 of 180 is "90"
Step-by-step explanation:
1 1/2 of 150 basically means 1 1/2 x 150, which equals to 225. 1/2 of 180 means 1/2 x 180 which is 90.
Hope this helps :)
2 short questions
Please help
Answer:
b c = 5.25p this is the answer
For the first problem shown the answer is number B.
It is B because we our always going up by $5.25
For our second one it isn't a function,
it isn't a function because it follows the same rule through-out it besides at the end of it.
Maya earned $225.oo at her job when she worked for 10 hours. What did she earn in one hour?
Solve the following system using the elimination method
−8y+6x=36
6x−y=15
Answer:
a
Step-by-step explanation:
Answer:
C. (2, -3)
Step-by-step explanation:
1.)Which of the following is a linear inequality in two variables?
a. 3x≤16
b. 3m-4m>5
c. 9q+4<12
d. 11+2t ≥ 3?
[tex] \large╚»★«╝ { \mathcal{Answer }}╚»★«╝[/tex]
The Correct option among them is B
[tex] \sf3m - 4n > 5[/tex]_____________________________________
The other options have one variable each, but according to the given question there should be two variables in the inequality.
The only choice satisfying the conditions is Option B
having two variables in it ( that is "m" and "n" )
[tex] ꧁ \: \large \frak{Eternal \: Being } \: ꧂[/tex]
It is 9 kilometers from Charlie's house to the nearest mailbox. How far is it in meters?
Be sure to include the correct unit in your answer.
Answer:
Step-by-step explanation:
9,000 meters