Determine the value of r so the line passing through each pair of points has the given slope
(6,3) (r,2), m=1/2

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.

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.

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.

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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).

Answer:

The answer is A! I hoped this helped! :3

Step-by-step explanation:

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.

The answer is c. sales tax varies by state.

[tex]\text{Distributive property:}\ a(b+c)=ab+ac\\\\4(y^5-2)=(4)(y^5)+(4)(-2)=\boxed{4y^5-8}[/tex]

The answer is A. yes, Y= 2x

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.

The length of each side of the square room with an area 225 ft² is 15 ft.

The area of a square is a product of it's any two sides or a product of diagonals divided by two. If side has a length a then diagonal is a√2.

The perimeter of a square is the sum of the lengths of all the sides.

Given, The area of a square room is 225 ft².

Assuming the length of each side is 'a' feet.

∴ The area of the square room is,

a×a =  225 ft².

a² = 225 ft².

a = ± 15 ft.

As length can not be negative +15 ft is the admissible value.

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Answer:

The equation of the line would be y = 6x + 2

Step-by-step explanation:

Since we have a slope (6) and a point (0,2) we can use them in point slope form. The slope goes in for m and the point goes in for (x1, y1).

y - y1 = m(x - x1)

y - 2 = 6(x - 0)

y - 2 = 6x

y = 6x + 2

standard form is y = mx+b

the correct answer is the 4th one y =4x+15

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.

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.

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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?

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

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.

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.