The evaluation of the values of the derivative of the function f, using Lagrange's theorem indicates that the true statements is the following option;
I and III anlyWhat does the Lagrange's Theorem highlights?Lagrange Theorem states that there is a point, c between points a and b of an arc such the slope at c is the same as the average slope of the arc between points a and b.
The evaluation of the statements in the question are as follows;
I. If f'(x) exists and is nonzero for all x, then f(1) is not equal to f(0)
If f'(x) exists and the value is non zero, then the slope is not infinite, and the sign of the slope of the function f(x) never changes, such that the as the input (x) value increases the output value decreases, when the slope is negative, or increases when the slope of f(x) is positive, therefore;
The value of f(1) is either larger than or lesser than the value of f(0), which proves that f(1) is not equal to f(0).
The first statement, I, is therefore, trueII. If f is differentiable for all x and f(-1) = f(1), then there is a number c, such that |c| < 1 and f'(c) = 0
The condition if f is differentiable for all x, then according to Lagrange's, mean value theorem, we get;
When f(|c| < 1) < f(1)
f'(c) < (f(1) - f(-1))/(1 - (-1)) = (f(1) - f(-1))/(2)
f'(c) < (f(1) - f(-1))/(1 - (-1)) = (f(1) - f(-1))/(2) = 0
f'(c) < 0
Similarly, when f(|c| < 1) > f(1), then, f'(c) < 0
Therefore, f'(c) ≠ 0
The statement II is not trueIII. If f'(c) = 0, then f has a local maximum or minimum at x = c.
The slope of the function f at point x = c is f'(c)
A local maximum or minimum is a point after which the slope of the graph changes sign, and the point at which the slope is zero as the change in the output variable is zero
Therefore, if f'(c) = 0, the graph has a local maximum or minimum at the point f = c
The true statements from the specified options, and based on the possible options in the question, obtained from a similar question posted online are;
I and III onlyThe possible question options are;
I and II only
I and III only
I only
II only
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A linear programming problem consists of a linear ____ to be to be maximized or minimized. It is subject to a set of constraints given in the form of linear equations or inequalities.
A linear programming problem consists of a linear function to be to be maximized or minimized. It is subject to a set of constraints given in the form of linear equations or inequalities.
Linear programming is the programming in which selecting the best possible choice from those available alternatives whose constraint function and the objective function can be referred to as the linear mathematical function.
Linear programming is used for the optimization of linear objective.
It is used as constrained optimization, from where the objective functions and constraints all are linear.
If a maximization linear programming problem consists of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables
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Determine the vertex and the zeros of the function: y = (x – 1)2 – 25. Question 1 options: A) Vertex: (–1,25); zeros: (–6,0), (4,0) B) Vertex: (–1,25); zeros: (6,0), (–4,0) C) Vertex: (1,–25); zeros: (6,0), (–4,0) D) Vertex: (1,–25); zeros: (–6,0), (4,0)
The vertex for the given function is (1, -25) and the zeros are (6, 0)(-4, 0).
What is a quadratic function?A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of the second degree.
Given y = (x - 1)² - 25
converting the equation in form of y = ax² + bx + c
y = x² + 1 - 2x - 25
y = x² - 2x - 24
vertex of a quadratic equation is given by (-b/2a, -D/4a)
b = -2, a = 1, c = -24
D = b² - 4ac
D = (-2)² - 4(-24)
D = 4 + 96
D = 100
vertex is, -b/2a = -(-2)/2 = 1
and -D/4a = -100/4 = -25
vertex are (1, -25)
and for zeros put y = 0
y = x² - 2x - 24
x² - 2x - 24 = 0
x² - 6x + 4x - 24 = 0
x(x - 6) + 4(x - 6) = 0
(x - 6)(x + 4) = 0
x = 6 and x = -4
zeros are (6, 0)(-4, 0)
Hence option C is correct.
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Suppose y varies directly with x. Write a direct variation equation that relates x and y. y=5 and x=2
The direct variation equation that relates x and y is 2y = 5x.
Direct variation:A type of proportionality known as "direct variation" occurs when one quantity directly changes in response to a change in another quantity.
This implies that if one quantity increases, the other quantity will also increase proportionately. Similarly, if one quantity decreases, the other amount also drops.
The symbol '∝' is used to represent the relation of Direct variation between two quantities.
Here we have
y varies directly with x
=> y ∝ x
=> y = kx ----(1)
where k is the proportionality constant
Given that x = 2 and y = 5
Now substitute given values in (1)
=> 5 = k(2)
=> 2k = 5
=> k = 5/2
Now substitute k = 5/2 in equation (1)
=> y = (5/2)x
=> 2y = 5x
Therefore
The direct variation equation that relates x and y is 2y = 5x.
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Writing and solving radical equations: Mastery Test
PLEASE HELP
The equation has 2 valid solutions; no extraneous solutions.
What are functions?A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.
What is factorization and radicals?Factorisation : The process of breaking or decomposing an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorisation or factoring.
Radicals :In the same way that addition is the opposite of subtraction in mathematics, a radical, also known as a root, is the opposite of an exponent. The square root, denoted by the symbol is the smallest radical.
The given equation is:
First, we determine the solutions
Add 5 to both sides
Square both sides
Expand
Collect like terms
Expand again
Factorize
Factor out x - 2
Split
The above values are valid values of x.
Hence, the equation has 2 valid solutions; no extraneous solutions
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Evaluate x^ 4 • x^-1 when x = 4.
A 64
B 1,024
C 1
1024
D. 64
When x = 4, x⁴ × x⁻¹ will be 1. So correct option is C.
What do you mean by exponential?In mathematics, an exponential function is a function in which the variable (usually denoted by "x") is in the exponent. The most common exponential function is f(x) = b^x, where b is a constant called the base of the exponential function. For example, f(x) = 2^x is an exponential function with base 2. The graph of an exponential function is a curve that increases or decreases at an ever-accelerating rate as x increases. The exponential function is useful in many areas of mathematics and science, including finance, population growth, and radioactive decay.
To evaluate x⁴ × x⁻¹ when x = 4, we first substitute 4 for x in the expression:
x⁴ × x⁻¹ = 4⁴ × 4⁻¹
Now we can simplify the exponent by multiplying 4 × -1 = -4
x⁴ × x⁻¹ = 4⁴ × 4⁻⁴
Now we can simplify the exponent by applying the rule (aᵇ) × (aⁿ) = a⁽ᵇ⁺ⁿ⁾
x⁴ × x⁻¹ = 4⁽⁴⁺⁽⁻⁴⁾⁾
We now have 4⁰ which is equal to 1, so the final answer is 1.
Therefore the answer is C. 1
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Ronnie loaned her older brother $1500 and he offered to pay her 4% interest in three years. How much money will Ronnie have in three years when her brother pays her back
Answer: 1560
Step-by-step explanation:1500*1.04=1560
Question 15
Divide £65 in the ratio 11:2
Questic
Divide
Answer:
55 and 10
I hope this helps
f (x) = log(x-1) +1
g(x) = log
What shifts occur for f(x)?
Answer:
f(x) shifts for log(DWX)
Step-by-step explanation:
HELP ASAP it is TIMED
Answer:24
Step-by-step explanation:
One factor of the function f(x) = x^3 - 6x^2 + 11x - 6 is (x-3)
The quotient of the polynomial division is x^2 - 3x + 2.
How to determine the quotient of the division?From the question, we have the following parameters that can be used in our computation:
f(x) = x^3 - 6x^2 + 11x - 6
Factor = (x-3)
One way to find the other factors of the function f(x) = x^3 - 6x^2 + 11x - 6 is to divide the polynomial by (x-3) and see what the quotient is.
To do this, we have
Quotient = f(x)/Factor
Substitute the known values in the above equation, so, we have the following representation
Quotient = (x^3 - 6x^2 + 11x - 6)/(x - 3)
Using a graphing calculator, we have
Quotient = x^2 - 3x + 2
Hence, the quotient is x^2 - 3x + 2
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Complete question
One factor of the function f(x) = x^3 - 6x^2 + 11x - 6 is (x-3)
Determine the quotient of the polynomial and the factor
Can’t figure this out
the data set shows the number of passengers on the flights leaving the columbus airport in the next hour. what is the median number of passengers on the flights? round answer to tenths place. if answer is a whole number then put a zero in your answer for the tenths place for it to be counted correct 39, 48. 25, 25, 53, 22, 70, 36
After rounding off the median of the given data would be 38.
What is the median?The median is the value that divides a data sample, a population, or a probability distribution's upper and lower halves in statistics and probability theory.
It could be referred to as "the middle" value for a data set.
The fundamental difference between the mean and the median when describing data is that the median is more representative of the "normal" value because it is not skewed by a tiny fraction of exceptionally big or small values.
So, calculate as follows:
22, 25, 25, 36, 39, 48, 53, 70
TErms are even: Average of two middle terms to get median.
Median = 36 + 39/2
Median = 37.5
Rounding off: 38
Therefore, after rounding off the median of the given data would be 38.
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draw a rectangular box with the origin and (2, 4, 5) as opposite vertices and with its faces paralle find the length of the diagonal of the box.
The length of the diagonal of the rectangular box with the origin O(0, 0, 0) and A(2, 4, 5) as opposite vertices is 6.708203932499369.
Let the origin be O(0, 0, 0) and A(2, 4, 5).
The length of the diagonal of the box can be calculated using the distance formula.
Diagonal = √((2-0)^2 + (4-0)^2 + (5-0)^2)
= √(4 + 16 + 25)
= √45
= 6.708203932499369
1. We are given the coordinates of two opposite vertices of a rectangular box, the origin O(0, 0, 0) and A(2, 4, 5).
2. To calculate the length of the diagonal of the box, we need to use the distance formula.
3. The length of the diagonal can be calculated as Diagonal = √((2-0)^2 + (4-0)^2 + (5-0)^2)
4. By substituting the values of the coordinates in the formula, we get Diagonal = √45
5. Simplifying the expression, we get Diagonal = 6.708203932499369.
The length of the diagonal of the rectangular box with the origin O(0, 0, 0) and A(2, 4, 5) as opposite vertices is 6.708203932499369.
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identifying congruent triangles determine if the two triangles are congruent. if they are, state how you know.
Triangle ABC and triangle XYZ are not congruent because they do not have the same side lengths or angles. To be congruent, two triangles must have the same side lengths and the same angles.
To determine if two triangles are congruent, we first need to check if they have the same side lengths. If the two triangles have the same side lengths, then we can move on to check if the angles of the two triangles are the same. If the two triangles have the same side lengths and the same angles, then we can conclude that the two triangles are congruent. If the two triangles do not have the same side lengths and the same angles, then we can conclude that the two triangles are not congruent. In the case of triangle ABC and triangle XYZ, they do not have the same side lengths or angles, so they are not congruent. To be congruent, two triangles must have the same side lengths and the same angles.
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i dropped a heavy cube the other day, creating a dent in my floor. the triangular hole in the surface of the floor has sides of lengths $68,$ $75,$ and $77.$ find the depth of the hole.
I recently dropped a large cube, leaving a dent in my floor. The floor's surface contains a triangular hole with sides that are $68, $75, and $77 long. The hole is about this deep: [tex]$d=12 \sqrt{6}[/tex]
What is meant by triangular?Three sides, three angles, and three vertices make up the closed, two-dimensional shape of a triangle. Polygons include triangles. Triangle ABC is seen in the above figure. Triangle examples. Sandwiches, traffic signals, clothing hangers, and a billiards rack are a few instances of triangles in everyday life.
To start, you can determine the sides of the tetrahedron whose apex corresponds to the dent's deepest point. Three of its sides are previously known to be $68,75,77, and since we have perpendicular planes, if these side lengths are x, y, and z, then
[tex]$x^2+y^2=68^2, x^2+z^2=75^2, y^2+z^2=77^2$[/tex] and these three equations solve to
[tex]$x=12 \sqrt{15}, y=4 \sqrt{154}, z=3 \sqrt{385}[/tex]
When the origin of a coordinate frame is located at the point where three perpendicular planes cross and the axes are placed along the edges, the equation for the plane of the triangle hole is
[tex]$\frac{x}{12 \sqrt{15}}+\frac{y}{4 \sqrt{154}}+\frac{z}{3 \sqrt{385}}=1[/tex]
As a result, the distance from the origin, which is the hole's deepest point, to its plane is
[tex]$d=\frac{1}{\sqrt{\frac{1}{144(15)}+\frac{1}{16(154)}+\frac{1}{9(385)}}}[/tex]
Then we get,
[tex]$d=12 \sqrt{6}[/tex]
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Answer the question in the photo
Answer:
16.89 cubic inches
Step-by-step explanation:
r = 2.25/2 = 1.125
h = 4.25
[tex]\pi[/tex] = 3.14
V = h[tex]\pi[/tex]r^2
V = (4.25)*(3.14)*(1.125)^2
V = 16.889765625
given the following venn diagram, where a and b are each represented by an oval: exam image which regions make up ( aexam imagebc )c?
The correct answer is U - II = I + III + IV.
In order to solve problems based on these sets, we can use a Venn diagram to depict the logical relationship between sets and their components. Although other closed figures like squares may be used, a Venn diagram commonly uses intersecting and non-intersecting circles to indicate the relationship between sets.
c represent the compliment of the particular state.
U is the union set
∩ represent intersection between the neighbouring sets
Therefore,
Bc = U - B = I + II
Bc∩A = Area common in Bc and A = II
(Bc∩A)c = Area in U but not in (Bc∩A) = U - II = I + III + IV
So, answer is d.
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Differentiation question please help step by step
Thus, we have shown that the equation [tex]\frac{d^2y}{dx^2} - 2tanx\frac{dy}{dx} = 0[/tex] is true using differentiation.
What is differentiation?A part from integration, differentiation is one of the two key principles. A technique for determining a function's derivative is differentiation.
Given y * cos x = e^x
By implicit differentiation, the first derivative is
-y * sin x + y' * cos x = e^x
By implicit differentiation, the second derivative is
-y cos x - y' sin x - y' sin x + y'' cos x
= e^x-y cos x - 2 * y' sin x + y'' * cos x
= e^x <--- combines the two like terms in the middle on left side
-y cos x - 2 * y' sin x + y'' * cos x = y * cos x <--- substitutes the original function y*cosx
= e^x-2y cos x - 2 * y' sin x + y'' * cos x = 0 <--- subtracts y*cos x from both sides
-2y - 2 * y' tan x + y'' = 0 <--- divides both sides by cos x
y'' - 2*tanx * y' - 2y = 0
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A ticket concert is on sale for $129. The sign sasy 14% off the original price. What was the original price?
Answer:
150
Step-by-step explanation:
x - .14x = 129
.86x = 129 Divide both sides by .86
x = 150
Which of the following represents all solutions (x,y) to the system of equations created by the linear equation and the quadratic equation y=x^2+9
All solutions to the system of equations created by the linear equation and the quadratic equation y=x^2+9 are (3,18), (-3,-18), (0,9), (-2,13), and (2,13).
To find all solutions to the system of equations created by the linear equation and the quadratic equation y=x^2+9, we need to solve for the intersection of the linear equation and the quadratic equation. The quadratic equation can be rewritten in the form of y=a(x-h)^2+k, where a, h, and k are constants. This equation can be set equal to the linear equation and solved using the quadratic formula. The solutions of this equation are the x-values that correspond to the intersection of the linear equation and the quadratic equation. To find the y-values, we need to substitute the x-values found in the quadratic equation. Therefore, all solutions to the system of equations created by the linear equation and the quadratic equation y=x^2+9 are (3,18), (-3,-18), (0,9), (-2,13), and (2,13).
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A convex octagon has interior angles with measures (x + 55)°, (3x +20)°, 4x°, (4x 10)°, (6x - 55)°, (3x +52)°, 3xº, and (2x + 30)°. Find the value of x.
Answer:
x = 38°
Step-by-step explanation:
sum of interior angle
=(n - 2 )180
=(8 - 2)180
=(6)180
=1080°.
the value of x
=(x + 55)° + (3x + 20)° + 4x° + (4x - 10)° + (6x - 55)° + (3x + 52)° + 3x° + (2x + 30)° = 1080°
= x + 3x + 4x + 4x + 6x + 3x + 3x + 2x + 55 + 20 - 10 - =55 + 52 + 30 = 1080.
=26x + 92 = 1080
=26x = 1080 - 92
=26x = 988
=26x/26 = 988/26
x = 38°
can someone pleaseeeee help me???????
Answer:
Hello the vertex point is (4, -6)
this parabola opens upwards because the coefficient of x is positive.
The answer is B, good luck!
If d represents the number of days taken to complete the work and p represents the number of people working, the equation that represents this situation is . If 10 people work on this job, they take days to complete it
Answer:
10d
Step-by-step explanation:
p = d
10p = ?
Cross multiply
(10pd)/p = 10d
Answer: The equation that represents the situation is d = k/p, where k is a constant.
This equation states that the number of days taken to complete the work (d) is inversely proportional to the number of people working on the job (p).
Given that 10 people work on the job and it takes them d days to complete it, we can substitute these values into the equation:
d = k/10
We don't know the value of k, so we can't find out the number of days it takes with 10 people working.
Step-by-step explanation:
Find the values of x and y
The value of x = 4 and y = 5.
What is congruence?
In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.
Here, we have
Given: triangles ABC and DEC are congruent, then:
3y + 1 = 4x
4y - 6 = 2x + 6
We need to solve the system of equations above.
Rearranging both equations:
-4x + 3y = -1 [1]
-2x + 4y = 12 [2]
Multiplying [2] by -2:
4x - 8y = -24 [3]
Adding [1] and [3]:
-5y = -25
Dividing by -5:
y = 5
Substituting in [1]:
-4x + 3(5) = -1
-4x + 15 = -1
Subtracting 15:
-4x = -16
Dividing by -4:
x = 4.
Hence, the value of x = 4 and y = 5.
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using the numbers 1-9, at most one time each, fill the blanks to make this equation true. find one solution for this equation.( ) = 81
The solution for this equation is (3¹)⁴=81.
The power of a number defines how many times to use the number in multiplication, Powers are also called Exponents or Indices.
If the power is positive, multiply the number by itself that many times. If the power is negative, multiply the number's reciprocal by itself that many times. If the power is zero, the result will always be 1.
For example, 8^2 could be called “8 to the power 2” “8 to the second power”, or simply “8 squared”.
A number, X, to the power of 2 is also referred to as X squared.
X is called the base number.
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A store sells a package of 25 trading cards for $5.25.
FIND SLOPE OF THE EQUATION: 5y - 3x = 12
A jet travels 480 miles in 5 hours. At this rate, how far could the jet fly in 8 hours? What is the rate of speed of the jet?
Answer:
Step-by-step explanation:
480 miles/5 hours=x miles/8 hours
x=480*8/5
x=768 miles(distance)
480/5=96 mph(speed)
Y=1+sinx/1-sinx find the derivative of this problem
The derivative of the problem is dy/dx = (2cosx)/(1-sinx)²
How to find the derivative of the function?The given function is Y=1+sinx/1-sinx
d/dx (u/v) = [u (dy/dx) - v (dv/dx) ] / v²or less formerly
(u/v) =[ [(v (du) = u(dv)] / v²
(Remember the rule in word; " vdu minus udv all over v squared)
So with Y=1+sinx/1-sinx then,
Let u = 1+sin x ⇒du/dx= cos x and
v = 1-sin x⇒dv/dx = -cos x
Then, [d/dx(u/v) = u(dy/dx) - u(dv/dx)] / v²
dy/dx = [(1-sinx)(cosx) - (1+sinx)(-cosx)] / (1-xin x)²
This implies that dy/dx =[ cosx - sin x +cosx +sin xcosx] / (1-sinx)²
Therefore, dy/dx = (2cos x)/(1-sinx)²
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a student eats a dinner containing
[tex]8.0 \times 10 { }^{6} [/tex]
joules of energy. he wishes to do an equivalent amount of work in a nearby gym by lifting a 60kg object. how many times must he raise the object to expand this much energy? assume he raises it a distance of 2.0m each time
The number of times a student must raise the object to expand the given energy level is 6800 times.
What is work and energy?Work is defined as transferring energy into an object so that there is some displacement. Energy is defined as the ability to do work. Work done is always the same. Energy can be of different types such as kinetic and potential energy.
Here, W=8.0×10⁶, m=60 kg, g=9.8 ms⁻² and h=2.0 m
We know that, W=mghn
Now, n=W/mgh
n=8.0×10⁶/(60×9.8×2)
= 6800 times
Therefore, the number of times a student must raise the object to expand the given energy level is 6800 times.
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