Answer:
No, because the z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual event, namely within 2 standard deviations of the mean of the sample means.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Unusual
If X is more than two standard deviations from the mean, x is considered unusual.
In this question:
[tex]\mu = 511, \sigma = 119, n = 55, s = \frac{119}{\sqrt{55}} = 16.046[/tex]
A random sample of 55 students from this school was selected, and the mean math SAT score was 528. Is the high school justified in its claim?
If Z is equal or greater than 2, the claim is justified.
Lets find Z.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{528 - 511}{16.046}[/tex]
[tex]Z = 1.06[/tex]
1.06 < 2, so 528 is not unusually high.
The answer is:
No, because the z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual event, namely within 2 standard deviations of the mean of the sample means.
The statement that could be made regarding the high school about the justification of its claim would be:
- No, because the z-score of Z = 1.06 is not unusual since it does not lie within the range of a usual event, namely within 2 standard deviations of the mean of the sample means.
Given that,
μ = 511
σ = 119
Sample(n) = 55
and
s = [tex]119/\sqrt{55}[/tex]
[tex]= 16.046[/tex]
As we know,
The claim of the high school could be valid and justified only when
[tex]Z > 2[/tex]
To find,
The value of Z
So,
[tex]Z = (X -[/tex] μ )/σ
by putting the values using Central Limit Theorem,
[tex]Z = (528 - 511)/16.046[/tex]
∵ [tex]Z = 1.06[/tex]
Since [tex]Z < 2[/tex], the claim is not justified.
Learn more about "Standard Deviation" here:
brainly.com/question/12402189
Length of a rod: Engineers on the Bay Bridge are measuring tower rods to find out if any rods have been corroded from salt water. There are rods on the east and west sides of the bridge span. One engineer plans to measure the length of an eastern rod 25 times and then calculate the average of the 25 measurements to estimate the true length of the eastern rod. A different engineer plans to measure the length of a western rod 20 times and then calculate the average of the 20 measurements to estimate the true length of the western rod.
Answer:
b. The engineer who weighed the rod 25 times.
Step-by-step explanation:
Hello!
Full text:
Length of a rod: Engineers on the Bay Bridge are measuring tower rods to find out if any rods have been corroded from salt water. There are rods on the east and west sides of the bridge span. One engineer plans to measure the length of an eastern rod 25 times and then calculate the average of the 25 measurements to estimate the true length of the eastern rod. A different engineer plans to measure the length of a western rod 20 times and then calculate the average of the 20 measurements to estimate the true length of the western rod.
Suppose the engineers construct a 90% confidence interval for the true length of their rods. Whose interval do you expect to be more precise (narrower)?
a. Both confidence intervals would be equally precise.
b. The engineer who weighed the rod 25 times.
c. The engineer who weighed the rod 20 times.
X₁: Length of an eastern rod of the Bay Bridge
n₁= 25
X₂: Length of a western rod of the Bay Bridge
n₂= 20
Both Engineers will use their samples to estimate the population average length of the rods using a 90% CI.
Assuming the standard normal distribution, the confidence interval will be centered in the estimated mean.
X[bar] ± [tex]Z_{1-\alpha /2}[/tex]*(σ/√n)
And the width is determined by the semi amplitude:
↓d= [tex]Z_{1-\alpha /2}[/tex]*(σ/√↑n)
As you can see the sample size has an indirect relationship with the semi amplitude of the interval. This means, when the sample size increases, the semi amplitude decreases, and if the sample size decreases, the semi amplitude increases. Naturally this is leaving all other elements of the equation constant, this means, using the same confidence level and the same population standard deviation.
Since the first engineer took the larger sample, he's confidence interval will be narrower and more accurate.
Hope this helps!
Write this number in expanded notation:178.25
Answer:
100+70+8+0.2+0.05. is the answer
Answer:
178.25 as a fraction is 178 1/4 or 713 / 4
Step-by-step explanation:
hope it works out !!
What’s the correct answer for this?
Answer:
57°
Step-by-step explanation:
According to theorem, "any two angles in the same segment of the circle are equal"
So,
m<BED = 57°
The sum of one and the product of 4 and a number x
Answer:
1 + 4x
Step-by-step explanation:
Let's break this down.
"The sum of one" means that something is being added to the number one:
1 +
"and" whatever comes after the word 'and' will be added to the number one
"the product of 4 and a number x" this means that the number four and the variable x are being multiplied:
4x
Put it together:
1 + 4x
Therefore, the expression is 1 + 4x.
A tree grows three feet per year. What happens to the growth of the
When the number of years increases, the number of feet decrea
When the number of years decreases, the number of feet stays
When the number of years increases, the number of feet increas
When the number of years decreases, the number of feet increa
Answer:
The answer is C :,)
Step-by-step explanation:
Answer:
The answer to your question is c
Step-by-step explanation:
Because the years have to increase for it to grow.
€ Practice
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Evaluation
Calculate simple interest
Question
Carolyn makes a deposit of $2,800 into a savings account. The bank calculates simple interest annually at a rate of 7.5%.
Interest is added every year on the anniversary of the initial deposit. How many years must Carolyn wait before her
investment exceeds $3,500? Give your answer in years. Do not include units in your answer.
Provide your answer below:
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SUBMIT
Content attribution
Answer:
4
Step-by-step explanation:
A=P0(1+rt).
We know that A=$3,500,P0=$2,800 and r=7.5%=0.075, so we can rearrange the formula for A to get an explicit expression for t:
A=P0(1+rt)⟹t=A−P0P0r,
and substituting the known quantities gives
r=$3,500−$2,800$2,800×0.075=3.33,
which means, since the interest is paid annually, that she must wait four years for the total investment to exceed $3,500.
A bag contains 4 green, 5 red, and 6 purple balls. The probability that all of them are red is?
Answer:
The percentage would be 20% (5x20=100)
Step-by-step explanation:
20sin^4 x power reduction
Answer:
Step-by-step explanation:
20 sin^4x
=5(4sin^4 x)
=5(2sin²x)²
=5(1-cos 2x)²
=5(1-4cos2x+cos²(2x))
=5[1-4cos(2x)+{1+cos (4x)}/2]
=5/2[2-8cos(2x)+1+cos(4x)]
=5/2[3-8cos (2x)+cos (4x)]
What’s the correct answer for this?
Answer:
C
Step-by-step explanation:
It's of the shape of a cone
Which table represents a relation that is not function
Please urgent
Florian ran 1.2 miles and walked 4.8 laps around the path at the park for a total distance of 3.6 miles. Which shows the correct equation and value of x, the distance of 1 lap around the path at the park? 3.6 x + 1.2 = 4.8; x = 1 mile 4.8 x + 1.2 = 3.6; x = 1 mile 3.6 x + 1.2 = 4.8; x = 0.5 mile 4.8 x + 1.2 = 3.6; x = 0.5 mile
Answer:
The correct answer would be D) 4.8x + 1.2 = 3.6; x = 0.5 mile
Step-by-step explanation:
This is because laps would be the dependent variable, so we know the number of them (4.8) would be multiplied by the variable (x). We also know that 1.2 is the constant. Now we can solve to make sure this is the right equation.
4.8x + 1.2 = 3.6
4.8x = 2.4
x = 0.5
Answer:
D) 4.8x + 1.2 = 3.6; x = 0.5 miles
Halley's birthday is on Friday. On Monday she receives 23 birthday cards. On Tuesday she receives 5 birthday cards. On Wednesday she receives 13 birthday cards. On Thursday she receives 17 birthday cards, and on Friday she receives 22 birthday cards. How many total birthday cards does Halley receive?
Answer:
80
Step-by-step explanation:
17+13=30
22+23+5=50
30+50=80
Answer:
80 cards
Step-by-step explanation:
23+5=28
13+17=30
28+30=58
58+22=80
Shape 1 and shape 2 are plotted on a coordinate plane. Which rigid transformation can you perform on shape 2 to show that shape 2 is congruent to shape 1?
What’s the correct answer for this?
Answer:
B and F
Step-by-step explanation:
When we'll slice a triangular prism, a square and triangle would be formed
Given that a = 5 , b = − 2 and c = − 2 work out 2 b − 3 a c
Answer:
26Step-by-step explanation:
[tex]a = 5 ,\\b = - 2 \\ c = - 2 \\ 2b - 3ac=?\\2(-2) -3(5)(-2)\\-4 +30\\= 26[/tex]
Answer:
-34
Step-by-step explanation:
2x-2-3(4)(-2)
which is -34
find the perimeter of this figure to the nearest hundredth use 3.14 to approximate pi P=?ft
Answer:
105.13ft^2
Step-by-step explanation:
[tex]A=lw\\=10*8\\=80ft^2[/tex]
Rectangle
[tex]A=\frac{1}{2} \pi r^2\\=\frac{1}{2\pi } 4^2\\=25.13[/tex]
Add both together
80+25.13
=105.13
Answer : 105.13
Step-by-step explanation:
lect the best answer for the question.
3
1. Find the value of y in the equation
-=8.
y-2
3
A. y = 2
8
B. y=-2-
3
8
5
C. y=-1
8
5
D. y =
loo
Tiffany is 140 miles away from Maggie. They are traveling towards each other. If Maggie travels 5 mph faster than Tiffany and they meet after 4 hours how fast was each traveling
Answer: Tiffany 15mph, Maggie 20mph
Step-by-step explanation:
Set up the equation 4((x+5) + x) = 140. x+5 represents how many miles Maggie covered in one hour. x represents how much Tiffany traveled in one hour. 140 is the number of miles in total. 4 is the number of hours in total.
Simplify the equation.
(x+5) + x = 35 Divide both sides by 4
2x+5 = 35 Combine like terms
2x = 30 Subtract 5 from both sides
x = 15 Divide both sides by 2
Tiffany traveled 15mph, while Maggie traveled 15+5=20mph.
Can someone plz help me solved this problem I need help ASAP plz help me! Will mark you as brainiest!
Answer:
8
Step-by-step explanation:
y²+by+16= (y+4)²
y²+by+16= y²+2*4*y+4²
y²+by+16= y²+8y+16
by=8y
b=8
Solve the inequality 2(4x-3)>-3(3x)+5
Answer:
48x>+2
Step-by-step explanation:
The base of a rectangular prism has an area of 24 square millimeters. The volume of the prism is 144 cubic millimeters. The shape is a cube. What is the height of the prism?
Answer:
height = 6 mm
Step-by-step explanation:
The prism is a rectangular prism. The base area of the prism is 24 mm². The volume of the prism is given as 144 mm³.
The height of the prism can be solved as follows.
Volume of the rectangular prism = Bh
where
B = base area
h = height
Volume = 144 mm³
B = 24 mm²
volume = Bh
144 = 24 × h
144 = 24h
divide both sides by 24
h = 144/24
h = 6 mm
Answer:
c
Step-by-step explanation:
edg 2022
We are planning on introducing a new internet device that should drastically reduce the amount of viruses on personal computers. We think the price should be $39.99, but are not sure on the percentage of people that would buy it. We do some research and find the following information; Studies from the 1930’s indicate that percentage should be between 30% and 40% Similar products were launched recently at a price of $4,000 and nobody bought it. A nationwide poll on this type of product and price was run earlier this year, with percentages running from 75% to 80%. We are going to conduct an additional focus group before we launch the product. What should the sample size be if we want a 95% CI to be within 5% of the actual value?
Answer:
The sample size required is 289.
Step-by-step explanation:
Let p be population proportion of people that would buy the product.
It is provided that the nationwide poll on this type of product and price was run earlier this year, with percentages running from 75% to 80%.
Assume that the sample proportion of people that would buy the product is, [tex]\hat p=0.75[/tex].
A 95% Confidence Interval is to be constructed with a margin of error of 5%.
We need to determine the sample size required for the 95% Confidence Interval to be within 5% of the actual value.
The formula to compute the margin of error for a (1 - α)% confidence interval of population proportion is:
[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The critical value of z for 95% confidence interval is,
z = 1.96.
Compute the sample size required as follows:
[tex]MOE=z_{\alpha/2}\times\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\ \sqrt{\hat p(1-\hat p)} }{MOE}]^{2}[/tex]
[tex]=[\frac{1.96\cdot \sqrt{0.75(1-0.75)} }{0.05}]^{2}\\\\=(16.9741)^{2}\\\\=288.12007081\\\\\approx 289[/tex]
Thus, the sample size required is 289.
Please answer this correctly
Answer:
1607.68 square miles
Step-by-step explanation:
use pi r squared
Answer:
Step-by-step explanation:
diameter = 64 miles
r =64/2 = 32 miles
Area of semicircle = πr²/2
= 3.14*32*32/2
= 1607.68 sq.miles
According to the Rational Root Theorem, Negative two-fifths is a potential rational root of which function?
f(x) = 4x4 – 7x2 + x + 25
f(x) = 9x4 – 7x2 + x + 10
f(x) = 10x4 – 7x2 + x + 9
f(x) = 25x4 – 7x2 + x + 4
Answer:
Neither expression satisfies the given rational root.Step-by-step explanation:
To find the right answer, we just need to replace the given root in each expression and see which one gives zero.
First expression.[tex]f(x)=4x^{4} -7x^{2} +x+25\\f(-\frac{2}{5})= 4(-\frac{2}{5})^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+25=\frac{64}{625}-\frac{28}{25} -\frac{2}{5} +25 \approx 23.58[/tex]
Second expression.[tex]f(x)=9x^{4}-7x^{2} +x+10=9(-\frac{2}{5} )^{4} -7(-\frac{2}{5} )^{2} +\frac{2}{5} +10 \approx 9.5[/tex]
Third expression.[tex]f(x)=10x^{4}-7x^{2} +x+9=10(-\frac{2}{5} )^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+9 \approx 7.7[/tex]
Fourth expression.[tex]f(x)=25x^{4}-7x^{2} +x+4=25(-\frac{2}{5} )^{4} -7(-\frac{2}{5})^{2} +(-\frac{2}{5})+4 \approx 3.12[/tex]
Therefore, neither expression satisfies the given rational root.
Answer:
D. f(x) = 25x^4 - 7x^2 + x + 4.
Step-by-step explanation:
The correct answer to your question is D.
Please answer this correctly
Answer:
Height of this missing bar would be 1
Step-by-step explanation:
Since there is 1 and only 1 quantity between 80-99.
Answer:
1
There is 1 number that is between 80 and 99 which is 99 so there should be 1 bar.
Step-by-step explanation:
A sample of carbon-12 has a mass of 6.00 g. How many atoms of carbon-12 are in the sample?
3.01 x 10^23
6.02 x 10^23
1.20 x 10^24
3.60 x 10^24
Answer:
The answer is 3.01 x 10’23
Step-by-step explanation:
I got the answer to wrong and guessed the first one and it was correct
Could someone please help me with the steps for this problem? Factor by grouping: w²+3w+w+3
Answer:
Please see steps below
Step-by-step explanation:
In order to factor by grouping, we divide the four terms given into two groups, and extract on each group any common factor we can.
In our example, we can select the terms: [tex]w^2[/tex] and [tex]w[/tex] as one of our groups, and [tex]3w[/tex] and 3 in the other group. Then we re-organize the expression as:
[tex](w^2+w)+(3x+3)[/tex]
Now we extract from the first binomial group, the factor [tex]w[/tex] as a common factor of both terms, and from the second group we extract the factor "3" as common factor of those two terms:
[tex](w^2+w)+(3x+3)\\w(w+1)+3(w+1)[/tex]
We notice now that after the extraction, we are left with two exactly equal binomial factors [tex](w+1)[/tex] that appeared in the first group and in the second group. We proceed then to extract it as common factor for the two groups:
[tex]w(w+1)+3(w+1)\\(w+1)(w+3)[/tex]
this last product of two binomials ([tex](w+1)\,(w+3)[/tex] is the result of factoring the original expression.
Can someone plz help me solved this problem I need help plz help me! Will mark you as brainiest!
Answer:
Step-by-step explanation:
let s note a and b
x = ap+b
we can write two equations
(1) 300=3a+b
(2) 450=1.5a+b
multiply by 2 the (2) we got
900 =3a+2b
minus (1) it gives
900 - 300 = 3a+2b-3a-b = b
so b = 600
and from (1) it gives 3a = 300-600 = -300
so a = -100
then
x=-100p+600
thanks
Zia is building a plastic model rocket that has the combined shape of a cone and a cylinder as shown. additionally, the cylinder has a hemisphere hollowed out of its bottom. the plastic for the cone weighs 1.4 grams per cubic centimeter and the plastic for the cylinder weights only 0.8 grams per cubic centimeter.
(a) the volume of plastic that remains in the cylinder after it has been hollowed out to the nearest cubic centimeter.
(b) what has a greater total weight, the plastic that makes up the cone or the plastic that makes up the cylinder after it has been hollowed out?
Answer:
226 cm^3
The mass of plastic used to make cylinder is greater
Step-by-step explanation:
Given:-
- The density of cone material, ρc = 1.4 g / cm^3
- The density of cylinder material, ρl = 0.8 g / cm^3
Solution:-
- To determine the volume of plastic that remains in the cylinder after gouging out a hemispherical amount of material.
- We will first consider a solid cylinder with length ( L = 10 cm ) and diameter ( d = 6 cm ). The volume of a cylinder is expressed as follows:
[tex]V_L =\pi \frac{d^2}{4} * L[/tex]
- Determine the volume of complete cylindrical body as follows:
[tex]V_L = \pi \frac{(6)^2}{4} * 10\\\\V_L = 90\pi cm^3\\[/tex]
- Where the volume of hemisphere with diameter ( d = 6 cm ) is given by:
[tex]V_h = \frac{\pi }{12}*d^3[/tex]
- Determine the volume of hemisphere gouged out as follows:
[tex]V_h = \frac{\pi }{12}*6^3\\\\V_h = 18\pi cm^3[/tex]
- Apply the principle of super-position and subtract the volume of hemisphere from the cylinder as follows to the nearest ( cm^3 ):
[tex]V = V_L - V_h\\\\V = 90\pi - 18\pi \\\\V = 226 cm^3[/tex]
Answer: The amount of volume that remains in the cylinder is 226 cm^3
- The volume of cone with base diameter ( d = 6 cm ) and height ( h = 5 cm ) is expressed as follows:
[tex]V_c = \frac{\pi }{12} *d^2 * h[/tex]
- Determine the volume of cone:
[tex]V_c = \frac{\pi }{12} *6^2 * 5\\\\V_c = 15\pi cm^3[/tex]
- The mass of plastic for the cylinder and the cone can be evaluated using their respective densities and volumes as follows:
[tex]m_i = p_i * V_i[/tex]
- The mass of plastic used to make the cylinder ( after removing hemispherical amount ) is:
[tex]m_L = p_L * V\\\\m_L = 0.8 * 226\\\\m_L = 180.8 g[/tex]
- Similarly the mass of plastic used to make the cone would be:
[tex]m_c = p_c * V_c\\\\m_c = 1.4 * 15\pi \\\\m_c = 65.973 g[/tex]
Answer: The total weight of the cylinder ( m_l = 180.8 g ) is greater than the total weight of the cone ( m_c = 66 g ).
The volume of the remaining plastic in the cylinder is large, which
makes the weight much larger than the weight of the cone.
Responses:
(a) Volume of the remaining plastic in the cylinder is 226 cm³(b) The weight of the cylinder is greater than the weight of the cone.How can the weight and volume be evaluated?Density of the plastic for the cone = 1.4 g/cm³
Density of the plastic used for the cylinder = 0.8 g/cm³
From a similar question, we have;
Height of the cylinder = 10 cm
Diameter of the cylinder = 6 cm
Height of the cone = 5 cm
(a) Radius of the cylinder, r = 6 cm ÷ 2 = 3 cm
Volume of a cylinder = π·r²·h
Volume of a hemisphere = [tex]\mathbf{\frac{2}{3}}[/tex] × π× r³
Volume of the cylinder after it has been hollowed out, V, is therefore;
[tex]V = \mathbf{\pi \times r^2 \times h - \frac{2}{3} \times \pi \times r^3}[/tex]Which gives;
[tex]V = \pi \times 3^2 \times 10 - \frac{2}{3} \times \pi \times 3^3 \approx \mathbf{ 226}[/tex]
Volume of the cylinder after it has been hollowed out, V ≈ 226 cm³(b) Volume of the cone = [tex]\mathbf{\frac{1}{3}}[/tex] × π × 3² × 5 ≈ 47.1
Mass of the cone = 47.1 cm³ × 1.4 g/cm³ ≈ 66 g
Mass of the hollowed cylinder ≈ 226 cm³ × 0.8 g/cm³ = 180.8 g
The mass and therefore, the weight of the plastic that makes up the hollowed cylinder is greater than the weight of the plastic that makes up the cone.Learn more about volume and density of solids here:
https://brainly.com/question/4773953
Find the absolute maximum and absolute minimum of the function f(x,y)=2x2−4x+y2−4y+1 on the closed triangular plate bounded by the lines x=0,y=2,y=2xin the first quadrant.
First check for the critical points of f by checking where the first-order derivatives vanish.
[tex]\dfrac{\partial f}{\partial x}=4x-4=0\implies x=1[/tex]
[tex]\dfrac{\partial f}{\partial y}=2y-4=0\implies y=2[/tex]
Notice how the point (1, 2) lies on the line y = 2x ; at this point, we get a value of f(1, 2) = -5 (MIN).
Next, check the points where the boundary lines intersect, which occurs at the points (0, 0), (0, 2), and (1, 2). We already checked the last one. We find f(0, 0) = 1 (MAX) and f(0, 2) = -3.
Now check on the boundary lines themselves. If x = 0, then
[tex]f(0,y)=y^2-4y+1=(y-2)^2-3[/tex]
which has a maximum value of -3 when y = 2 (so we get the same critical point as before).
If y = 2, then
[tex]f(x, 2)=2x^2-4x-3=2(x-1)^2-5[/tex]
with a maximum of -5 when x = 1.
If y = 2x, then
[tex]f(x,2x)=6x^2-12x+1=6(x-1)^2-5[/tex]
with the same maximum of -5 when x = 1.
This question is based on the absolute maximum and absolute minimum.
We get this by differentiating the terms.
Given:
f(x,y) = [tex]2x^{2} - 4x + y^2 - 4y +1[/tex], bounded by the lines x=0,y=2,y=2x in the first quadrant,bounded by the lines x=0,y=2,y=2x in the first quadrant.
We need to determined the absolute maximum and absolute minimum of the function.
Now, partial differentiating wrt x and y.
[tex]\dfrac{\partial f}{ \partial x} = 4x -4 = 0 \Rightarrow x= 1 \\\dfrac{\partial f}{ \partial y} = 2y - 4 = 0 \Rightarrow y = 2[/tex]
Now, point (1, 2) lies on the line y = 2x ; at this point, we get a value of
f(1, 2) = -5 (MIN).
Next, check the points where the boundary lines intersect, which occurs at the points (0, 0), (0, 2), and (1, 2).
Now, find f(0, 0) = 1 (MAX) and f(0, 2) = -3.
Now check on the boundary lines themselves.
If x = 0, then we get,
[tex]f(0,y) = y^2 - 4y +1 = ( y-2)^2 -3\\[/tex]
which has a maximum value of -3 when y = 2 (so we get the same critical point as before).
If y = 2, then we get,
f(x,2) = [tex]2x^2-4x -3 = 2(x-1)^2 -5[/tex] with a maximum of -5 when x = 1.
If y = 2x, then we get,
f(x,2x) = [tex]6x^2 -12x +1 = 6(x-1)^2 -5[/tex] with the same maximum of -5 when x = 1.
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