Barry wants to surprise his family with a vacation to the Bahamas. So he buys plane tickets with cash in hand and they all go.
Barry wants to surprise his family with a vacation to the Bahamas. So he uses all his cash, then uses his credit card to purchase the plane tickets.
Find the critical numbers of the function f(x) = x6(x − 1)5 what does the first derivative test tell you that the second derivative test does not?
The critical numbers of [tex]\( f(x) = x^6 (x-1)^5 \)[/tex] are [tex]\( x = 0, 1, \frac{5}{6} \)[/tex]. The first derivative test determines maxima/minima by sign change, while the second derivative test confirms concavity but not explicit categorization.
1. Calculate the first derivative [tex]\( f'(x) \)[/tex]:
Use the product rule and simplify:
[tex]\[ f(x) = x^6 (x-1)^5 \] \[ f'(x) = 6x^5 (x-1)^5 + x^6 \cdot 5(x-1)^4 \] \[ f'(x) = 6x^5 (x-1)^4 [x + 5(x-1)] \] \[ f'(x) = 6x^5 (x-1)^4 (6x - 5) \][/tex]
2. Find critical points:
Critical points occur where [tex]\( f'(x) = 0 \)[/tex] or is undefined.
Set [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ 6x^5 (x-1)^4 (6x - 5) = 0 \][/tex]
This equation is zero when:
[tex]\( 6x^5 = 0 \) -- \( x = 0 \)\\ \( (x-1)^4 = 0 \) -- \( x = 1 \)\\ \( 6x - 5 = 0 \) -- \( x = \frac{5}{6} \)[/tex]
So, the critical numbers are [tex]\( x = 0, 1, \frac{5}{6} \)[/tex].
3. First Derivative Test vs. Second Derivative Test:
- First Derivative Test: This test uses the sign of the derivative ( f'(x) ) to determine the nature of critical points:
- If [tex]\( f'(x) > 0 \) to \( f'(x) < 0 \)[/tex] at x, then ( f(x) ) has a local maximum at x.
- If [tex]\( f'(x) < 0 \) to \( f'(x) > 0 \)[/tex] at x, then ( f(x) ) has a local minimum at x.
- Second Derivative Test: This test uses the sign of the second derivative [tex]\( f''(x) \)[/tex]:
- If [tex]\( f''(x) > 0 \), \( f(x) \)[/tex] has a local minimum at x.
- If [tex]\( f''(x) < 0 \), \( f(x) \)[/tex] has a local maximum at x.
4. Difference between the tests:
- The first derivative test provides information about the nature (whether maximum or minimum) of critical points based on the sign change of the derivative.
- The second derivative test provides more precise information by confirming the concavity of the function around the critical point, but it does not give explicit information about whether the critical point is a maximum or minimum (it only confirms the direction of concavity).
Determine whether the formula describes y as a function of x. Explain your reasoning. y=-3x^2-7x-6
If your brother is 12 years old and you are 10 years old. How old will you have to be to be half your brother's age?
Answer:
6
Step-by-step explanation:
Joana is 4 years older than her brother. Which of the following expressions shows the sum if their ages if b represents joanas brothers age?
What is the expanded form of the expression 3 sigma i^3 i=0
Which of the above charts does NOT accurately reflect public goods and services?
A.
chart A
B.
chart B
C.
chart C
D.
chart D
Answer:
The answer is A! I hoped this helped! :3
Step-by-step explanation:
What is the quotient? 2m+4/8÷m+2/6
Answer:
3/2 (D on Edge)
Woodland Bicycles makes two models of off-road bicycles: the Explorer, which sells for $250, and the Grande Expedition, which sells for $350. Both models use the same frame, but the painting and assembly time required for the Explorer is 2 hours, while the time is 3 hours for the Grande Expedition. There are 375 frames and 450 hours of labor available for production. How many of each model should be produced to maximize revenue?
To maximize revenue, we need to determine the number of each model of off-road bicycles that should be produced based on the given constraints. The revenue generated from selling each model is multiplied by the number of units sold. We set up a system of equations to represent the constraints and objective function, and then use graphical methods or linear programming algorithms to find the optimal solution.
Explanation:To maximize revenue, we need to determine the number of each model of off-road bicycles that should be produced. Let's assume x represents the number of Explorer bicycles and y represents the number of Grande Expedition bicycles.
The revenue generated from selling the Explorer bicycles is 250x, while the revenue from selling the Grande Expedition bicycles is 350y.
We are given that the painting and assembly time required for the Explorer is 2 hours, so the total time used for producing the Explorer bicycles is 2x hours. Similarly, the total time used for producing the Grande Expedition bicycles is 3y hours.
We also know that there are 375 frames and 450 hours of labor available. Each bicycle requires one frame and a certain number of labor hours, so the total number of frames used is x + y, and the total labor hours used is 2x + 3y.
We can set up the following system of equations to represent the constraints:
x + y ≤ 375 (due to the limited number of frames)
2x + 3y ≤ 450 (due to the limited number of labor hours)
To maximize revenue, we need to maximize the objective function R = 250x + 350y. Since we are dealing with a linear programming problem, we can use graphical methods or linear programming algorithms to find the optimal solution.
2,) What is the solution of the system?
Use the substitution method.
y+6=4x
8x−2y=14
A) The only solution is (14, 12)
B)The only solution is (12, 14)
C)There are an infinite number of solutions.
D)There is no solution.
Say that you are about to make a major purchase and would like to save money on sales tax. Which of these is a viable way to reduce the amount you will pay on sales tax?
a.
Sales tax is a tax like any other, so you can claim additional dependents and get a deduction on it.
b.
Businesses are allowed to charge a 1% fee for automatically calculating sales tax for you, so you should calculate the sales tax out yourself.
c.
Sales tax varies by state, so you can travel to a state with a lower tax rate and make your purchase there.
d.
If you buy an item on credit, the credit bureau must pay the sales tax instead of you, so you should make your purchase using a credit card.
The answer is c. sales tax varies by state.
If ABC and DEF are supplementary, then ABC and DEF form a linear pair
The statement is true. If two angles are supplementary meaning their sum is 180 degrees, they form a linear pair because in a linear pair too, the sum of the two angles is 180 degrees.
Explanation:The question is related to the concept of angles in geometry. When angles are supplementary, it means that the sum of the two angles is 180 degrees. In a linear pair, two adjacent angles form a straight line, which also sums up to 180 degrees.
So, if angles ABC and DEF are supplementary, meaning ABC + DEF = 180 degrees, they indeed form a linear pair. Because in a linear pair too, the sum of the two angles is 180 degrees. Therefore, the statement 'If ABC and DEF are supplementary, then ABC and DEF form a linear pair' is true.
Learn more about Supplementary Angles and Linear Pair here:https://brainly.com/question/34190098
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The correct question is given below:
If ABC and DEF are supplementary, then ABC and DEF form a linear pair. True or False?
If f(x) = 2x2 + 4, find f(3). (1 point) 22 16 −0.5 40
Does the density of an iron bar depend on the length of the bar
HELP Use the properties of sigma notation and the summation formulas to evaluate. See attached
in what quadrant is an angle of 1.75 found
Use the dustributive property to express 16+48?
PLEASE HELP!!
To the nearest square unit, what is the area of the regular heptagon shown below?
A. 2772 square units
B. 1572 square units
C. 792 square units
D. 5545 square units
Answer:
1572 Square Units
Step-by-step explanation:
First, find the perimeter by multiplying 20.8 * 7 which is 145.6. Then multiply the perimeter with the apothem. After that, you divide by 2.
So, 145.6*21.6/2= 1572.
What is the value of the expression?
(52)2+334
Enter your answer in the box.
Answer:
438
Step-by-step explanation:
(52*2)+334
104+334=
438
The value of the numerical expression (52)² + 334 will be 3,038.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
The numerical expression is given below.
⇒ (52)² + 334
Simplify the numerical expression, then we have
⇒ (52)² + 334
⇒ 2,704 + 334
⇒ 3,038
The value of the numerical expression (52)² + 334 will be 3,038.
More about the Algebra link is given below.
https://brainly.com/question/953809
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How Do I do proofs for proving triangles?
A prepaid cell phone charges a preset number of minutes to use text messaging. The graph represents y, the number of minutes used for x, the number of text messages sent and received. Is there a direct variation? Which equation represents the relationship?
A .Yes, y = 2x. B. Yes, y = 20x. C. No, y = x + 10. D. No, y = x + 20.
The answer is A. yes, Y= 2x
Octagon ABCDEFGH and its dilation, octagon A'B'C'D'E'F'G'H', are shown on the coordinate plane below:
Octagon ABCDEFGH with ordered pairs at A at 3, 1, at B 3, negative 1, C 1, negative 3, D negative 1, negative 3, E negative 3, negative 1, F negative 3, 1, G negative 1, 3, H 1, 3. Octagon A prime B prime C prime D prime E prime F prime G prime H prime with ordered pairs A prime 6, 2, at B prime 6, negative 2, at C prime 2, negative 6, at D prime negative 2, negative 6, at E prime negative 6, negative 2, at F prime negative 6, 2, at G prime negative 2, 6, at H prime 2, 6
If the center of dilation is at the origin, by what scale factor was octagon ABCDEFGH dilated?
one half
2
4
one quarter
octagon ABCDEFGH was dilated by a scale factor of 2. Therefore, option 2 is the correct answer.
To find the scale factor by which octagon ABCDEFGH was dilated, we can compare the corresponding side lengths of the original octagon and the dilated octagon.
Let's take one side, for example, side AB. The length of side AB in the original octagon is the distance between points A(3, 1) and B(3, -1), which is 2 units.
The length of side A'B' in the dilated octagon is the distance between points A'(6, 2) and B'(6, -2), which is also 4 units.
By comparing the side lengths, we can see that side A'B' is twice as long as side AB.
Therefore, the scale factor by which octagon ABCDEFGH was dilated is 2.
In conclusion, octagon ABCDEFGH was dilated by a scale factor of 2.
Therefore, option 2 is the correct answer.
The octagon ABCDEFGH was dilated by a scale factor of 2. This means each coordinate in the original octagon was multiplied by 2 to obtain the corresponding point in A'B'C'D'E'F'G'H'.
The scale factor of dilation between octagon ABCDEFGH and its dilated counterpart A'B'C'D'E'F'G'H' can be determined by comparing corresponding side lengths. Taking one side as an example, the original octagon has a side connecting points A(3, 1) and B(3, -1) with a length of 2 units (vertical distance). In the dilated octagon, the corresponding side connecting A'(6, 2) and B'(6, -2) has a length of 4 units (vertical distance). The scale factor (k) is then calculated as the ratio of the length in the dilated octagon to the length in the original octagon: [tex]\( k = \frac{4}{2} = 2 \)[/tex].
Therefore, octagon ABCDEFGH was dilated by a scale factor of 2 from its original position to A'B'C'D'E'F'G'H'. This indicates that the coordinates of each point in the original octagon were multiplied by 2 to obtain the corresponding points in the dilated octagon. The scale factor, in this case, is 2, signifying a doubling in size during the dilation process.
The lines described by y = (a + 12)x + 3 and y = 4ax are parallel. what is the value of a
The lines are parallel if and only if the values of their slopes are equal. In a standard form of the equation of the line, the slope is the numerical coefficient of x. Given this condition, we determine the value of a by equation a + 12 and 4a.
Slope = a + 12 = 4a
The value of a from the equation is 4.
Answer: 4
Experts/acee plz helppp
standard form is y = mx+b
the correct answer is the 4th one y =4x+15
distribute 4(y5-2 i neeed help
[tex]\text{Distributive property:}\ a(b+c)=ab+ac\\\\4(y^5-2)=(4)(y^5)+(4)(-2)=\boxed{4y^5-8}[/tex]
Mike can stitch 7 shirts in 42 hours. He can stitch 1 shirt in hours, and in 1 hour he can stitch of a shirt.
Answer:
He can Stitch 1 shirt in 6 Hours And in Hour He Can Stitch 6/10 Of a Shirt
Step-by-step explanation:
This Answer Is Correct Because He Can Stitch 7 Shirts In 42 Hours Which Is Wrong.
Simplify 5x + 8y + 4x
A.17x
B.9x + 8y
C.5x + 12y
D.17xy
For which to pairs of function is (f•g) (x) =x
Answer:
[tex]f(x)=\frac{2}{x}[/tex] and [tex]g(x)=\frac{2}{x}[/tex]
Step-by-step explanation:
To prove that we have to demonstrate [tex](f*g)(x)=x[/tex]
This proof is about a composition of functions, where we have to enter one function inside another, in this case, [tex]g(x)[/tex] goes inside [tex]f(x)[/tex]. So if results in [tex]x[/tex], then will be proved.
[tex](f*g)(x) = \frac{2}{\frac{2}{x} }[/tex]
As you can see, the composition is to replace [tex]g(x)[/tex] for the variable of [tex]f(x)[/tex].
Solving the expression:
[tex](f*g)(x) = \frac{2}{\frac{2}{x} }=\frac{2x}{2}=x[/tex]
After replacing and applying the composition, we have [tex]x[/tex] as a result. Therefore, we can say that the pair of functions of option B is the answer, because they satisfy the expression given.
Multiply. 3 ⋅ 3/5 whats the answer
Write a function rule to represent the situation. the total cost C for pp pound of beef if each pound costs $4.01
What is 4.898979486 rounded to the nearest 10?