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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On a coordinate plane, the coordinates of vertices R and T for a polygon are R(-6,2) and T(5,2). what is the length of side RT of the polygon?
Answer:
YoOoooo its 7 units
Step-by-step explanation:
the binomial expansion of (x-2y)^3 is ____.
A. x^3-x^2y+12xy^2-8y^3
B. x^3-3x^2y+3xy^2-y^3
C. x^3-3x^2y+3xy^2+y^3
D. 2^3(x^3-3x^2y+3xy^2-y^3)
A= x^3−6x^2y+12xy^2−8y^3
Money off coupons have been circulated to 300 households. Only 2/5 of these were redeemed (used) in the local supermarket to get a free shampoo. What fraction of coupons were unused?
(PS: Please write how you go the answer)
What's the proper name of the quadrilateral below?
Need the steps to get the answer 0.264 that is on my answer key textbook
If ab = 8 and a^2+b^2=16, then what is the value of (a+b)^2?
how many students were reading during study time?
Dennis surveyed 30 students to determine their favorite fall sports. Of the students surveyed, 15 preferred football, 6 preferred soccer, and the rest preferred volleyball. Which is a correct statement about the results from the survey?
Find the general solution of the differential equation y'' - 2y' - 3y = -3te^(-t)
If u = 3 inches, v = 4 inches, w = 7 inches, x = 6 inches, y = 8 inches, and z = 5 inches, what is the area of the object?
A.60 square inches
B.66 square inches
C.63 square inches
D.56 square inches
The area of the object will be 63 square inches. Then the correct option is C.
What is Geometry?It deals with the size of geometry, region, and density of the different forms both 2D and 3D.
If u = 3 inches, v = 4 inches, w = 7 inches, x = 6 inches, y = 8 inches, and z = 5 inches.
All the dimensions are shown in the figure.
Then the area of the geometry is the combination of two rectangle and one triangle.
Area = area of rectangle 1 + area of rectangle 2 + area of triangle 3
Area = 5 x 8 + 4 x 5 + (1/2) x 3 x 2
Area = 40 + 20 + 3
Area = 63 square inches
Then the correct option is C.
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which represents the solution(s) of the equation x2=289?
a. x equals the square root of 17
b. x equals positive square root of 17
c. x equals negative square root of 289
d. x equals positive square root of 289
Answer:
The answer are the options C and D
x equals negative square root of [tex]289[/tex]
x equals positive square root of [tex]289[/tex]
Step-by-step explanation:
we have
[tex]x^{2} =289[/tex]
Take square root both sides
[tex]x=(+/-)\sqrt{289}[/tex]
therefore
the solutions are
[tex]x=+\sqrt{289}[/tex]
[tex]x=-\sqrt{289}[/tex]
Give a description of a3/4 turn of the minute hand on a clock face
how do you write one half the difference of X and 4?
Write the point-slope form of the equation of the line that passes through the points (-8, 2) and (1, -4). Include your work in your final answer. Type your answer in the box provided or use the upload option to submit your solution.
What is the product of (–0.9)(–2.4)?
–3.3
–2.16
2.16
3.3
Two negatives terms multiplied together make a positive, so the product of (–0.9)(–2.4) will be equal to 2.16
What is the fundamental principle of multiplication?If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
WE know that Two negatives multiplied together make a positive, so the two negatives would cancel each other out when multiplying. This would basically make the solution.
WE have to find the product of (–0.9)(–2.4)
it’s simply (–0.9) x (–2.4) = 2.16
Now from the first option 3.3 is definitely not less than 2.4, so the answer would be positive 2.16
Hence, the product of (–0.9)(–2.4) will be equal to 2.16
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what is the area of this
The total area of the figure, calculated by summing the areas of three rectangles, is 81 square yards.
The figure in the image is composed of three rectangles. To find the total area, we calculate the area of each rectangle and then sum them up.
The first rectangle has a length of 13 yards and a width of 5 yards. Using the formula for the area of a rectangle, which is length times width, we get an area of 13 yd * 5 yd = 65 square yards.
The second rectangle extends upwards with a length of 2 yards and a width of 5 yards. Its area is 2 yd * 5 yd = 10 square yards.
The third rectangle extends to the right with dimensions of 2 yd by 3 yd. Its area is 2 yd * 3 yd = 6 square yards.
Adding these areas together, we get a total area of 65 sq yd + 10 sq yd + 6 sq yd = 81 square yards.
The question probable may be:
Find the area of the figure. (sides meet at right angles.)
The area of the piecewise rectangular figure is 59 square yards.
How the area is determined:
The first rectangle:
Length = 13 yards
Width = 5 yards
Area of the first rectangle = 65 square yards (13 x 5)
The second rectangle:
Length = 13 yards
Width = 2 yards
Area of the second rectangle = 6 square yards (3 x 2)
Since the second rectangle reduces the first rectangle, we subtract:
Area of the piecewise rectangle = 59 square yards (65 - 6)
Thus, we can conclude that the area of the piecewise rectangular figure is 59 square yards.
If Sandy can afford car payments of $260 per month for 3 years, what is the price of a car that she can afford now? Assume an interest rate of 7.8 percent.
The diameter of a circle is 6 yards. What is the circle's area? d=6 yd Use 3.14 for .
Answer: 28.26 square yards
Step-by-step explanation:
The diameter is twice the radius, so if the diameter is 6, the radius is 3.
[tex]A=\pi(3^{2})=9\pi \approx 9(3.14)=\boxed{28.26 \text{ yd}^{2}}[/tex]
Jake needs to put gas in the delivery van. Regular gasoline is $3.20 per gallon. He only had a $ 20 bill on him. How many gallons can he buy
if the height of a rectangular prism is doubled then the surface area of the prism is and I need work to and can you please help me and thank you
find (f*f)(3) f(x)=x^2-x
To find (f * f)(3) for the function f(x) = x² - x, we calculate f(3) to get 6, and then square it to get 36.
The question is asking to calculate (f * f)(3) given the function f(x) = x² - x. To do this, we find the value of f(x) for x=3 and then square that result. First, let's compute f(3).
f(3) = (3)² - (3) = 9 - 3 = 6.
Now, we find (f * f)(3), which means f(3) * f(3).
(f * f)(3) = f(3) * f(3) = 6 * 6 = 36.
Therefore, the solution to (f * f)(3) is 36.
After a concert for 1,600 people, only 4 people said they would not go to see the band again. What percent of the people who went to the concert said they would not go to see the band again?
The answer you are looking for is 0.25%.
To find this, you must first divide 4 by 1,600, to find the decimal amount of people who said they wouldn't go again. This would give you 0.0025, which you then multiply by 100 to make a percentage. Thus making 0.25%, the correct answer.
I hope this helps!
Answer:
0.25 percent
Step-by-step explanation:
Table Sasha wants to know her average grade on her math tests. She earned a score of 70 on two tests, an 80 on one test, and a 90 on two tests. What is Sasha’s mean score? 70 75 80 90 This question is for 79 points
Helppp experrts/ace/geniuses
solve the inequality 2× - 5 > 7
A drawbridge in Jacksonville, FL, allows an average of 22 boats per day to pass. What does the point (4,88) on the graph mean in this situation?
Answer: This shows, 'After 4 days, 88 boats will have passed under the bridge'.
Explanation:
Since, According to the question, Jacksonville allows an average of 22 boats per day to pass.
Thus we can say that number of boats is dependent on days and both are varying.
And, a function is the type of y=f(x), where x is independent variable while y is dependent variable.
So, we can take x for representing number of days and y for representing number of boats.
Thus, according to the question, y= 22x.
if we take 4 days i.e. x=4 then y=22×4=88
⇒It takes 4 days for an average of 88 boats.
Which is showing by (4,88) on the given graph.
Find the length and width of a rectangle that has the given perimeter and a maximum area. perimeter: 244 meters
The sum of 4 consecutive odd numbers is 40. What is the 2nd number in this sequence?
The length of a garden is 5m greater than the width. the area is 66m^2. Find the dimensions of the garden.
Daisy designs 30 cards in a day and Lisa designs 12 cards in a day how many more cards does daisy design than Lisa
To find out how many more cards Daisy designs than Lisa, subtract the number of cards Lisa designs from the number of cards Daisy designs. Daisy designs 18 more cards than Lisa.
Explanation:To find out how many more cards Daisy designs than Lisa, we need to subtract the number of cards Lisa designs from the number of cards Daisy designs.
Daisy designs 30 cards in a day and Lisa designs 12 cards in a day.
Therefore, Daisy designs 30 - 12 = 18 more cards than Lisa.
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