Answer:
See the explanation.
Step-by-step explanation:
Given function [tex]f(x)=x^3-x^2-6x+2[/tex]
And the interval [tex][0,3][/tex]
According to Rolle's Theorem
Let [tex]f(x)[/tex] be differentiable on the open interval [tex](a,b)[/tex] and continuous on the closed interval [tex][a,b][/tex]. Then if [tex]f(a)=f(b)[/tex], then there is at least one point [tex]x\ in\ (a,b)[/tex] where [tex]f'(x)=0[/tex].
So,
[tex]f(0)=0^3-0^2-6\times0+2\\\\f(0)=2\\\\Similarly\\\\f(3)=3^3-3^2-6\times3+2\\\\f(3)=27-9-18+2\\\\f(3)=2\\\\We\ can\ see\ f(0)=f(3)\\\\Now,\\\\f'(x)=\frac{d}{dx}(x^3-x^2-6x+2)\\\\f'(x)=3x^2-2x-6\\\\put\ f'(x)=0\\\\3x^2-2x-6=0\\[/tex]
We will find the value of [tex]x[/tex] for which [tex]f'(x)[/tex] became zero.
[tex]If\ ax^2+bx+c=0\\\\Then,\ x_{1}=\frac{-b+\sqrt{b^2-4ac}}{2a}\\ \\And\ x_{2}=\frac{-b-\sqrt{b^2-4ac}}{2a}\\\\f'(x)=3x^2-2x+6=0\\a=3,\ b=-2,\ c=6\\\\\ x_{1}=\frac{-(-2)+\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2+\sqrt{76}}{6}=\frac{2+2\sqrt{19}}{6}\\\\x_{1}=\frac{1+\sqrt{19}}{3}\\\\x_{2}=\frac{-(-2)-\sqrt{(-2)^2-4\times3\times(-6)}}{2\times3}\\\\x_{1}=\frac{2-\sqrt{76}}{6}=\frac{2-2\sqrt{19}}{6}\\\\x_{2}=\frac{1-\sqrt{19}}{3}[/tex]
We can see
[tex]x_{1}=\frac{1+\sqrt{19}}{3}=1.786\\\\and\ 1.786\ is\ in\ [0,3][/tex]
There is at least one point [tex]1.786\ in\ (0,3)[/tex] where [tex]f'(x)=0[/tex].
Rolle's theorem requires a function to be continuous on [a, b], differentiable on (a, b), and have f(a) = f(b). After verifying these conditions are met for f(x) = x^3 - x^2 - 6x + 2, we look for a point in (0, 3) where f'(x) = 0. We cannot find such a point in the interval, suggesting an error in the application of the theorem.
To verify Rolle's theorem for the function f(x) = x^3 - x^2 - 6x + 2 in the interval [0,3], we need to ensure that the function meets the criteria stated by the theorem:
The function f must be continuous on the closed interval [a, b].
The function f must be differentiable on the open interval (a, b).
The function f must satisfy f(a) = f(b).
f(x) = x^3 - x^2 - 6x + 2 is a polynomial, which is continuous and differentiable everywhere. Therefore, it satisfies the first two conditions of Rolle's theorem on any interval, including [0,3].
Now we need to check the third condition:
f(0) = (0)^3 - (0)^2 - 6(0) + 2 = 2
f(3) = (3)^3 - (3)^2 - 6(3) + 2 = 2
Since f(0) = f(3), the third condition is also met. Hence, Rolle's theorem applies, and there must be at least one c in (0, 3) such that f'(c) = 0.
To find c, we take the derivative of f(x):
f'(x) = 3x^2 - 2x - 6
Set f'(x) equal to zero and solve for x:
3x^2 - 2x - 6 = 0
Factors into (3x + 2)(x - 3) = 0
So x can be either x = -2/3 or x = 3. Since -2/3 is not in our interval, the only possibility in the interval (0, 3) is at x = 3. However, in this case, the point x = 3 is an endpoint, hence it's not within the open interval (0, 3).
Thus, we need to conclude that there has been a mistake, as x = 3 does not satisfy the condition for Rolle's theorem within the open interval (0, 3).
What is the least number of drops of red, blue, and yellow pigment that Ava
can add to a gallon of white paint so that the custom color consists of 25%
red, 35% blue, and 40% yellow pigment?
Answer:
5 red, 7 blue and 8 yellow
Step-by-step explanation:
Just make a ratio and simplify
Red : Blue : Yellow
25 : 35 : 40
All divisible by 5
5 : 7 : 8
These dont have a common factor so this is the simplest ratio
Therefore the minimum drops Ava can add to a gallon of white paint to have that ratio are 5 red drops, 7 blue drops and 8 yellow drops.
A radio station asks its listeners to call in their opinion regarding the closing of fire stations in the city. Identify the sampling method used, and explain why this sample could be biased.
Answer:
A convenience sample is used. The sample could be biased because it limits the population to listeners of that radio station at a certain time. Callers may be more likely to have a strong opinion on the issue.
Answer:
A convenience sample is used. The sample could be biased because it limits the population to listeners of that radio station at a certain time. Callers may be more likely to have a strong opinion on the issue.
Step-by-step explanation:
It limits population to listeners of that radio station at only a particular time
7 + n / 6
plz help me out
42+n/6
Step-by-step explanation:
I don’t have brainly plus but I actually need help understanding this..
Answer:
[tex]x^2=3x(x-2)[/tex]
Step-by-step explanation:
Let length of square be "x"
According to problem,
Length of Rectangle is 3 times, so
Length of Rectangle = 3x
Also,
Width of rectangle is 2 units less than length of square, so
Width of Rectangle = x - 2
Now, area of square is Side * Side
and
area of rectangle = length * width
Hence,
Area of Square = [tex]x^2[/tex]
Area of Rectangle = [tex](3x)(x-2)[/tex]
If they are equal, we can write:
[tex]x^2=3x(x-2)[/tex]
Brianna's family spent $134 on 2 adult tickets and 3 youth tickets at an amusement park. Max's family spent $146 on 3 adults tickets and 2 youth tickets. What is the price of a youth ticket?
The price of 1 youth ticket is $ 22
Solution:
Let "y" be the price of 1 youth ticket
Let "a" be the price of 1 adult ticket
To find: price of 1 youth ticket
Brianna's family spent $134 on 2 adult tickets and 3 youth tickets at an amusement park
So we can frame a equation as:
2 adult tickets x price of 1 adult ticket + 3 youth tickets x price of 1 youth ticket = 134
[tex]2 \times a + 3 \times y = 134[/tex]
2a + 3y = 134 ---- eqn 1
Max's family spent $146 on 3 adults tickets and 2 youth tickets
So we can frame a equation as:
3 adult tickets x price of 1 adult ticket + 2 youth tickets x price of 1 youth ticket = 146
[tex]3 \times a + 2 \times y = 146[/tex]
3a + 2y = 146 ----- eqn 2
Let us solve eqn 1 and eqn 2 to find values of "y"
Multiply eqn 1 by 3
6a + 9y = 402 ---- eqn 3
Multiply eqn 2 by 2
6a + 4y = 292 ----- eqn 4
Subtract eqn 4 from eqn 3
6a + 9y = 402
6a + 4y = 292
( - ) -------------------
5y = 110
y = 22
Thus the price of 1 youth ticket is $ 22
PLEASE HELP SOLVE
1.-8p - 7 ≥ - 5 - 5p - 4p p ≥
2.r - 1 ≤ - 8 + 2r r ≥
3.a + 6 < 6 + 2a a >
4.16 - 4x + 1 + 5 < 1 + x + 2x x >
Answer:
1) p > or =3
2)7< or = r
3)a<0
4) 3< x
Step-by-step explanation:
follow on the attached picture
(4x+7) and [5(x-4)} what is the value of x
Answer:
The value of x is 27
Complete question and statement:
(4x + 7) = [5(x - 4)} what is the value of x?
Source: Previous question that can be found at brainly
Step-by-step explanation:
Let's solve for x, (4x + 7) = [5(x - 4)]
(4x + 7) = [5x - 20]
4x + 7 = 5x - 20
4x - 5x = - 20 - 7 (Subtracting - 5x and - 7 at both sides)
- x = - 27
x = 27 (Multiplying by -1 at both sides)
The value of x is 27
by what would you multiply each side of the equation ax=27 to solve x
Answer:
Step-by-step explanation:
ax= 27........I would normally solve this by division....but it says multiply....so multiply both sides by 1/a
example :
3x = 27.....multiply both sides by 1/3
1/3(3/1)x = 27 * 1/3
x = 27/3 which reduces to 9
u have to multiply by the reciprocal of a to get rid of a
e. A semicircular window with a radius of
36 inches allows light through an area of how
many square feet? You may use a calculator
if one is available. Round your answer to the
nearest square foot.
Answer:
14 sq. ft [tex](14 ft^{2})[/tex]
Step-by-step explanation:
First, we know that the area of a circle is [tex]\\ A_{circle} =\pi * r^{2}[/tex], where r is the radius of the circle, and [tex]\pi[/tex] is the ratio of the length of a circle's circumference to its diameter, that is, [tex]\pi[/tex]=3.1415926535 ... . But we have here a semicircle.
So, the area of a semicircle is [tex]\\ A_{semicircle} = \frac{1}{2} (\pi * r^{2})[/tex].
We also know that 1 feet = 12 inches, so 36 inches are:
[tex]\\ \frac{1ft}{12inches} * 36 inches = 3 ft[/tex]
Then, the area of the semicircular window is:
[tex]\\ A_{semicircular-window} = \frac{1}{2} (\pi * 3^{2})[/tex] ≈ 14.1372.
Well, rounding the answer to the nearest square foot, we have that the semicircular window allows light through an area of 14 sq. ft [tex] (14 ft^{2})[/tex].
-5q- -9q - -14=2 solve for q
Answer:
q = − 1 \6
Step-by-step explanation:
q = − 0.1 6 ¯ 6
Percy said that any real number for k would cause the system of equations to have no solution. Explain the error in Percy’s statement.
6x + 4y = 14,
3x + 2y = k
Answer:
Except k=7, any real number for k would cause the system of equations to have no solution.
Step-by-step explanation:
In general a system of equations can be represented as ax+by=c and dx+ey=f. In order this system of equations to have NO SOLUTIONS a/d=b/a≠c/f. In our example a=6, b=4, c=14, d=3, e=2 and f=k. To apply the formula above, 6/3=4/2≠14/k. Hence k≠7. It can be concluded that except k=7, any real number for k would cause the system of equations to have no solutions.
Just for information, if k=7 the system will have infinitely many solutions.
The error in Percy's statement is that, the system of equations would have infinite many solutions when k = 7
The system of equations is given as:
6x + 4y = 14,
3x + 2y = k
Multiply the second equation by 2
[tex]3x + 2y = k \to 6x + 4y = 2k[/tex]
Subtract the new equation, from the first equation
[tex]6x - 6x + 4y - 4y =14 -2k[/tex]
[tex]0=14 -2k[/tex]
Collect like terms
[tex]2k = 14[/tex]
Solve for k
[tex]k = 7[/tex]
The above means that:
The system of equations would have infinite many solutions when k = 7
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What is the answer ?
Answer:
31415 92653
Step-by-step explanation:
The result of the integral is π. The first 30 digits of π are ...
3.14159 26535 89793 23846 26433 8327 ...
_____
Pi is a transcendental number. Not only is it irrational, but it is not the root of any polynomial with rational real coefficients. It is not a repeating decimal.
what is the solution to the equation below? Round your answer to two decimal places. 4*e^x=12.76
Answer:
[tex]x=1.16[/tex]
Step-by-step explanation:
we have
[tex]4(e^x)=12.76[/tex]
Solve for x
That means ---> Isolate the variable x
Divide by 4 both sides
[tex](e^x)=3.19[/tex]
Apply ln both sides
[tex]ln[(e^x)]=ln(3.19)[/tex]
Apply property of logarithms (power rule) left side
[tex](x)ln(e)=ln(3.19)[/tex]
Remember that
[tex]ln(e)=1[/tex]
[tex]x=ln(3.19)[/tex]
[tex]x=1.16[/tex]
Answer:1.16
Step-by-step explanation:
this is correct
Y=7 what type of line
Answer:
Horizontal line or the technical name: Zero slope.
Step-by-step explanation:
When graphed, the line comes out horizontally, and horizontal lines are called zero slopes.
Step-by-step explanation:
y = aa - any real number
This is the equation of the horizontal line passing through the points in the form (x, a), where x is any real number.
A slope of a horizontal line is equal 0.
y = 7It's a horizontal line passing through the points in the form (x, 7).
Look at the picture.
Can someone plzzz help me with number 8 plzz show work plzzzz
Answer:
x = 7, y = 28
Step-by-step explanation:
We're given that ABCD is a rectangle, so we know that all sides opposite from each others are equal in length. This also means that each part of the criss-cross-X thing inside the rectangle are equal in length to each others as well. (AE = BE = DE = CE).
AC = 100 from the given picture, which means half of that from AE would be 50. Since DE is the same length as AE, we can set x²+1 = 50 and solve for x. We get x = 7.
To solve for y, we know that opposite sides of a rectangle are of the same length. So AD = BC.
The question already gave us the equation for those sides so we just need to set them equal to each other.
2y + 5 = 3y - 23
sovle for it and you get y = 28.
How do I Graph y= –73x+2 .
Answer:
See attachment
Step-by-step explanation:
You can obtain any two points on the graph of [tex]y=-73x+2[/tex] and use it to draw its graph.
When x=0, [tex]y=-73*0+2=2[/tex]
So you plot (0,2)
When x=1, [tex]y=-73*1+2=-71[/tex]
You again plot (1,-71).
With a straight edge you can now draw a straight line through the two points.
See attachment
Solve. 4y – 3 = 5y + 2
–5
–3
5
45
Answer:
y = - 5
Step-by-step explanation:
Given
4y - 3 = 5y + 2 ( subtract 5y from both sides )
- y - 3 = 2 ( add 3 to both sides )
- y = 5 ( multiply both sides by - 1 )
y = - 5
How many times could 6 go into 49
6 goes into 49 8 times with a remainder of 1.
Make x the subject:
1. ax + 2d = 2cx + 5b
Answer:
x = [tex]\frac{5b-2d}{a-2c}[/tex]
Step-by-step explanation:
Given
ax + 2d = 2cx + 5b (subtract 2d from both sides )
ax = 2cx + 5b - 2d ( subtract 2cx from both sides )
ax - 2cx = 5b - 2d ← factor out x from each term on the left side
x(a - 2c) = 5b - 2d ← divide both sides by (a - 2c)
x = [tex]\frac{5b-2d}{a-2c}[/tex]
The value of x is x = (5b - 2d )/(a - 2c)
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Given that;
ax + 2d = 2cx + 5b
Now subtract 2d from both sides;
ax = 2cx + 5b - 2d
Then subtract 2cx from both sides;
ax - 2cx = 5b - 2d
Now factor out x from each term on the left side
x(a - 2c) = 5b - 2d
Now divide both sides by (a - 2c)
x = (5b - 2d )/(a - 2c)
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Solve the system by elimination. −7x+y=−19 −2x+3y=−19
Answer:
x=2, y=-5. (2, -5).
Step-by-step explanation:
-7x+y=-19
-2x+3y=-19
--------------------
-3(-7x+y)=-3(-19)
-2x+3y=-19
----------------------
21x-3y=57
-2x+3y=-19
-------------------
19x=38
x=38/19
x=2
-7(2)+y=-19
y=-19-(-14)
y=-19+14=-5
A bulletin board has a perimeter of 200 inches. Which equation expresses the area A(w) of the bulletin board as a function of its width, w?
A. A(w) = 100w - w2
B. A(w) = 200w - w2
C. A(w) = w2 – 100w
D. A(w)=w2 - 200w
The equation that expresses the area A(w) of the bulletin board as a function of its width w is A(w) = w².
Explanation:To express the area A(w) of the bulletin board as a function of its width, we need to find the equation that relates the width to the area. Since the perimeter of the bulletin board is given as 200 inches, we can use the formula for the perimeter of a rectangle: P = 2w + 2l, where w is the width and l is the length. Since the bulletin board is assumed to be a rectangle, the length is equal to the width. We can rewrite the formula as 2w + 2w = 200, which simplifies to 4w = 200. Solving for w, we find w = 50.
Now that we know the width, we can express the area A(w) as a function of w. Since the area of a rectangle is given by the formula A = length * width, and the length is equal to the width in this case, we have A(w) = w * w = w².
The correct equation that expresses the area A(w) of the bulletin board as a function of its width w is A(w) = w², which corresponds to option D.
Write as many fractions as you can that are greater than one and less than two, with a denominator of three
Step-by-step explanation:
hwjxh ejdowbvrnrks n.v eveid
Is this statement true or false?
Slope is calculated by dividing the change in rise by the change in run.
Answer:
true
Step-by-step explanation:
Answer:
true
Step-by-step explanation:
What is the greatest number of y-intercepts a line can have? Explain
Answer:
infinity
Step-by-step explanation: it keeps going
You purchased 6 items at the 2-Dollar Store costing $1.98, $.59, $2.99, $.99, $2.49, and $.49. What is the mean price of your purchased items?
Answer:
$1.58
Step-by-step explanation:
To find the mean price, add up all the prices and divide by the number of items. The mean price of your purchased items is $1.59.
Explanation:To find the mean price of the purchased items, you need to add up all the prices and then divide by the number of items. In this case, you purchased 6 items. So, you add $1.98 + $0.59 + $2.99 + $0.99 + $2.49 + $0.49 = $9.53. Then, you divide $9.53 by 6 to get the mean price which is $1.59.
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Please help me answer and explain number 55
Answer:
D
Step-by-step explanation:
An operation of two real numbers is defined by the rule
[tex]a\bigotimes b=b^a+2ab[/tex]
Calculate [tex]2\bigotimes (1\bigotimes 3).[/tex] First evaluate the expression in brackets:
[tex]1\bigotimes 3=1^3+2\cdot 1\cdot 3=1+6=7[/tex]
Now,
[tex]2\bigotimes (1\bigotimes 3)=2\bigotimes 7=2^7+2\cdot 2\cdot 7=128+28=156[/tex]
The total cost to go horseback riding is x dollars per hour plus a $3 fee for renting a helmet. Write an expression to represent the cost, in dollars, for 5 people to go horseback riding for 2 hours each.
The expression to represent the cost, in dollars, for 5 people to go horseback riding for 2 hours each is $ (15 + 10x)
Solution:
Given that The total cost to go horseback riding is "x" dollars per hour plus a $3 fee for renting a helmet.
total cost = $ 3 + "x" dollars per hour
To find: expression to represent the cost, in dollars, for 5 people to go horseback riding for 2 hours each.
Let us first find the total cost for 1 person to horse ride for 2 hours
total cost for 1 person = $ 3 + 2x
Thus to find total cost for 5 people to go horseback riding for 2 hours each is found by multiplying the above expression by 5
Total cost for 5 people = 5 x total cost for 1 person
[tex]\text{ Total cost for 5 people } = 5 \times (3 + 2x)[/tex]
[tex]\text{ Total cost for 5 people } = 15 + 10x[/tex]
Thus the expression to represent the cost, in dollars, for 5 people to go horseback riding for 2 hours each is $ (15 + 10x)
To the nearest tenth, what is the distance between the point (10, -11) and (-1, -5)
The distance between the points (10, -11) and (-1, -5) is approximately 12.5 units.
Explanation:The distance between two points can be found using the Distance Formula.
The formula is:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given points (10, -11) and (-1, -5), we can plug the values into the formula:
distance = sqrt((-1 - 10)^2 + (-5 - (-11))^2)
distance = sqrt((-11)^2 + (6)^2)
distance = sqrt(121 + 36)
distance = sqrt(157)
To the nearest tenth, the distance is approximately 12.5 units.
simplify each expression 1/3^-3
Answer:
27
Step-by-step explanation:
Use exponent laws
1/3^-3 = 3^3 = 27
What is the value of x?
D x= 2.25
DX = 11.25
0 x = 13
0 x = 22
Answer:
[tex]x=22[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The exterior angle is the sum of the 2 opposite interior angles of a triangle
so
In this problem
[tex](6x+1)^o=79^o+(2x+10)^o[/tex]
solve for x
Combine like terms
[tex]6x+1=89+2x[/tex]
Group terms
[tex]6x-2x=89-1[/tex]
[tex]4x=88[/tex]
divide by 4 both sides
[tex]x=22[/tex]