Let f be continuous on the interval I = [a, b] and let c be an interior point of I. Assume that f is differentiable on (a, c) and (c, b). If there is a neighborhood (c − δ, c + δ) ⊆ I such that f ′ (x) ≤ 0 for c − δ < x < c and f ′ (x) ≥ 0 for c < x < c + δ. Prove that, f has a relative minimum at c
Answers
To prove that f has a relative minimum at c, we can use the First Derivative Test. The First Derivative Test states that if a function is differentiable on an interval and the derivative changes sign from negative to positive at a point within that interval, then that point is a relative minimum.
Given that f is continuous on the interval I = [a, b], differentiable on (a, c) and (c, b), and that f'(x) ≤ 0 for c − δ < x < c and f'(x) ≥ 0 for c < x < c + δ, we can proceed with the proof:
Consider the left neighborhood of c, (c - δ, c). Since f is differentiable on (a, c), we can apply the Mean Value Theorem (MVT) on this interval. According to the MVT, there exists a point d between a and c such that f'(d) = (f(c) - f(a))/(c - a).
Since f'(x) ≤ 0 for c − δ < x < c, it follows that f'(d) ≤ 0. This implies that f(c) - f(a) ≤ 0.
Consider the right neighborhood of c, (c, c + δ). Applying the MVT again, there exists a point e between c and b such that f'(e) = (f(b) - f(c))/(b - c).
Since f'(x) ≥ 0 for c < x < c + δ, it follows that f'(e) ≥ 0. This implies that f(b) - f(c) ≥ 0.
Combining the inequalities from steps 2 and 4, we have f(b) - f(c) ≥ 0 ≥ f(c) - f(a).
Since f(b) - f(c) ≥ 0 ≥ f(c) - f(a), it follows that f(b) ≥ f(c) ≥ f(a).
Therefore, f(c) is a relative minimum because it is smaller than or equal to the function values at both endpoints of the interval I = [a, b].
In conclusion, based on the given conditions and the application of the First Derivative Test, we have shown that f has a relative minimum at c.
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Related Questions
Write an explicit formula for the nth term of this sequence:
6, 11, 16, 21, . . .
1. an=5n+11
2. an=5n+1
3. an=21-5n
4. an=6n+1
Answers
Answer:
2)an = 5n + 1
Step-by-step explanation:
This is an arithmetic series.
6 , 11 , 16, 21 ,.....
First term = \(a_{1}\) = 6
Common difference = d = second term - first term
= 11 - 6 = 5
nth term = \(a_{1}+(n-1)*d\)
= 6 + (n - 1) * 5
= 6 + n * 5 - 1*5
= 6 +5n - 5
= 6 - 5 + 5n
= 1 + 5n
Translate this sentence into an equation.
58 is the difference of Helena's height and 9.
Use the variable h to represent Helena's height.
Answers
The algebraic form of the stentence; 58 is the difference of Helen's height and 9 is 58 = h - 9.
What is the height of Helen?Given the statement; 58 is the difference of Helen's height and 9.
Let the variable 'h' represent Helen's height.
We can translate the sentence into an equation.
Using h as Helen's height;
598= the difference of Helen's height and 9
58 = h - 9
Hence, the stentence 58 is the difference of Helen's height and 9 in an algebraic form is 58 = h - 9
We can go on and solve for h.
58 = h - 9
h - 9 = 58
Add 9 to both sides
h - 9 + 9 = 58 + 9
h = 58 + 9
h = 67
Therefore, Helen's height is 67.
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What is the value of the expression 2 3/4 (-8.5)
A.-16 3-4
B.-16 3/8
C.-23 3/4
D.-23 3/8
Answers
Answer:
D. -23 3/8
Step-by-step explanation:
When doing the math correctly you should get -23.375 or -23 3/8
help me please. i have been doing this for a long time
Answers
Answer:
I am deeply sorry.
Step-by-step explanation:
I am really sorry about how long it takes you to do "this". My best advice is to take a break and think it out slowly.
What is the estimated standard error for a sample of n = 9 scores with ss = 288?
Answers
The estimated standard error for a sample of n = 9 scores with ss = 288 is 2. Among your choices, it should be B; s2 = 36 and sM = 2.
Hope that helps!
a cylindircal container can be made by using two circles for the ends and a rectangle which wraps round to form the body. what is the largest possible cylinder
Answers
Answer:
Step-by-step explanation:
Answer:
Difference= $11,280
Step-by-step explanation:
Giving the following information:
Manager at Shop & Save Grocers= Hourly wages $16.50 per hour
Assistant Manager at Unbelievable Toys= Monthly wages $3,800
Cashier at 7-12 Convenience Store= Yearly wages $43,000
First, we need to calculate the yearly wages, and then compare the results:
Shop & Save Grocers= (16.5*40)*52= $34,320
Unbelievable Toys= 3,800*12= $45,600
The highest paying job is at Unbelievable Toys.
Difference= 45,600 - 34,320= $11,280
Rosa walked 3 miles in 60 minutes. At that rate, how long would it take her to walk 5 miles?
Answers
Answer:
100 minutes (an hour and 40 minutes)
Step-by-step explanation: divide 60 by 3 to get 20 and multiply 20 by 5 to get 100.
ASAP!
The table represents a quadratic function C(t).
t C(t)
−2 1
−1 4
0 5
1 4
2 1
What is the equation of C(t)?
C(t) = −(t − 5)2
C(t) = (t − 5)2
C(t) = −t2 + 5
C(t) = t2 + 5
Answers
The equation of the Quadratic function C(t) based on the given table is C(t) = -t^2 + 5.
The equation of the quadratic function C(t) based on the given table, we need to find the pattern and form of the equation that fits the given values.
Looking at the table, we can see that the values of C(t) vary as t changes. By examining the corresponding values of t and C(t), we can observe that the function appears to be symmetric and reaches its maximum value at t = 0.
From the table, we can see that when t = 0, C(t) = 5. This suggests that the vertex of the quadratic function is located at the point (0, 5). Since the function is symmetric, the vertex form of the quadratic equation can be written as:
C(t) = a(t - h)^2 + k,
where (h, k) represents the vertex of the parabola.
Given that the vertex is (0, 5), we can substitute these values into the equation:
C(t) = a(t - 0)^2 + 5,
C(t) = a(t^2) + 5.
To find the value of 'a', we can substitute the coordinates of another point from the table into the equation. Let's use the point (1, 4):
4 = a(1^2) + 5,
4 = a + 5,
a = -1.
Substituting the value of 'a' back into the equation, we have:
C(t) = -t^2 + 5.
Therefore, the equation of C(t) is C(t) = -t^2 + 5.
In conclusion, the equation of the quadratic function C(t) based on the given table is C(t) = -t^2 + 5.
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a sample of 900 computer chips revealed that 46% of the chips do not fail in the first 1000 hours of their use. the company's promotional literature states that 44% of the chips do not fail in the first 1000 hours of their use. the quality control manager wants to test the claim that the actual percentage that do not fail is different from the stated percentage. find the value of the test statistic. round your answer to two decimal places.
Answers
Therefore, the value of the test statistic is 2.02 (rounded to two decimal places).
To test the claim that the actual percentage of computer chips that do not fail in the first 1000 hours of their use is different from the stated percentage of 44%, we can use a hypothesis test with the following null and alternative hypotheses:
Null hypothesis: The proportion of computer chips that do not fail in the first 1000 hours of their use is equal to 44%.
Alternative hypothesis: The proportion of computer chips that do not fail in the first 1000 hours of their use is not equal to 44%.
We can use a normal approximation to the binomial distribution to calculate the test statistic:
z = (p - P) / √(P(1-P)/n)
where:
p = sample proportion = 0.46
P = hypothesized proportion = 0.44
n = sample size = 900
Plugging in the values, we get:
z = (0.46 - 0.44) / √(0.44*0.56/900)
z = 2.02
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Select the non−linear function from the list of the following:
Select one:
a.
xy + 1 / x = 7
b.
5x + 6y = − 2 / 3
c.
x + 3y = − 2
d.
− 4x − 5y = − 3
Answers
Answer:
a. xy + 1/x = 7
Step-by-step explanation:
The first equation has terms with degrees 2 and -1. A linear equation has variable terms only with degree 1.
The nonlinear equation is ...
xy + 1/x = 7
an unbiased coin is tossed 13 times. in how many ways can the coin land tails either exactly 6 times or exactly 3 times?
Answers
The number of ways an unbiased coin lands tails either exactly 6 times or exactly 3 times when tossed 13 times is 59.9.
Binomial distribution gives a summary of the number of trials as well as observations as each trial has the same probability of attaining one particular value.
The unbiased coin is tossed 13 times.
p(6) = ₁₃C₆ × (0.5)⁶ × (0.5)¹³ ⁻ ⁶
p(3) = ₁₃C₃ × (0.5)³ × (0.5)¹³ ⁻ ³
P = p(9) + p(3)
P = ₁₃C₆ × (0.5)⁶ × (0.5)¹³ ⁻ ⁶ + ₁₃C₃ × (0.5)³ × (0.5)¹³ ⁻ ³
P = ₁₃C₆ × (0.5)¹³ + ₁₃C₃ × (0.5)¹³
P = (0.5)¹³ [₁₃C₆ + ₁₃C₃]
P = (0.5)¹³ × [ 13! / 6!( 7!) + 13! / 3!( 10! ) ]
P = (0.5)¹³ × [ 13! / 6!( 7!) + 13! / 3!( 10! ) ]
P = (0.5)¹³ [ 13 × 11 × 2 × 3 × 2 + 13 × 2 × 11 ]
P = 0.00012207031 × 2002
P = 0.24438476062
Hence, the unbiased coin can land tails either exactly 6 times or exactly 3 times in 59.9 ways.
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What’s a prime factors of 12?
Answers
Answer: 2 and 3.
Step-by-step explanation:
The numbers 2 and 3 are the only prime factors of 12, but a prime factorization of 12 must list the 2 twice — 2 × 2 × 3 (or 22 × 3), because 2 × 3, by itself, doesn't make 12.
(a) What 3 by 3 matrix E₁₃ will add row 3 to row 1? (b) What matrix adds row 1 to row 3 and at the same time adds row 3 to row 1? (c) What matrix adds row 1 to row 3 and then adds row 3 to row 1?
Answers
The process of adding the matrix is explained blow.
Matrices are powerful tools used to analyze data and solve complex equations.
A 3x3 matrix, or E₁₃, is a matrix composed of 3 rows and 3 columns. It can be used to perform various operations, including adding two rows together.
(a) To answer the first question, a 3x3 matrix E₁₃ can add row 3 to row 1 by performing a row operation. This involves taking the second row and adding it to the first, thus adding the values on each row together.
(b) To answer the second question, a 3x3 matrix E₁₃ can add row 1 to row 3 and at the same time add row 3 to row 1 by performing a row exchange operation. This results in a 3x3 matrix where each value in the first row is equal to the corresponding value in the third row.
(c) To answer the third question, a 3x3 matrix E₁₃ can add row 1 to row 3 and then add row 3 to row 1 by performing two separate row operation and then taking the third row and adding it to the first, thus adding the values of each row together again.
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Turn into fractions the following periodic decimal numbers. Present each fraction in its simplest form.
Answers
Choose the option that best describes the limiting values of T and a under the conditions given. Choose the option that best describes the limiting values of and under the conditions given.
A T=0 and a=0
B T=[infinity] and a=0
C T=mg and a=0
D T=[infinity] and a=g
E T=0 and a=[infinity]
F T=[infinity] and a=[infinity]
Answers
Option C best describes the limiting values of T and a under the conditions given. In this case, T represents tension and a represents acceleration.
Without the specific conditions mentioned, it is impossible to determine the exact limiting values of T and a. However, certain options can be ruled out based on common sense and physical laws. For example, option C (T=mg and a=0) is not possible as the tension in a string cannot be equal to the weight of an object. Option E (T=0 and a=[infinity]) is also not possible as a mass cannot have zero tension and infinite acceleration.
Based on these eliminations, the most reasonable options are A (T=0 and a=0) and D (T=[infinity] and a=g). In the former case, the object is not moving and there is no tension in the string. In the latter case, the object is in free fall and the tension in the string is negligible compared to the weight of the object.
However, it is important to note that the exact limiting values of T and a will depend on the specific conditions of the scenario, such as the mass of the object and the angle of the string.
Option C best describes the limiting values of T and a under the conditions given. In this case, T represents tension and a represents acceleration. When T=mg and a=0, it means that the tension in the system is equal to the gravitational force acting on the mass (mg) and the system is in equilibrium with no acceleration. This is a common scenario when an object is hanging from a rope or cable and not moving. The other options do not represent stable or realistic conditions for a physical system.
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I WILL GIVE YOU BRAINLIEST FOR THIS ONE
Answers
Anyone know whats
90-209
+1-23+2046.
Answers
Answer:
1905
Step-by-step explanation:
How do you make comparisons between an equation and a table?.
Answers
Overall, equations provide a mathematical representation and understanding of relationships, while tables present data points in a structured format.
When comparing an equation and a table, you can examine their characteristics, purpose, and the information they provide. Here are some points to consider when making comparisons:
Representation:
An equation represents a mathematical relationship between variables using symbols and mathematical operations. A table presents data or values in a structured format, typically organized into rows and columns.
Communication of Information:
Equations provide a concise and general representation of a relationship, allowing for predictions, calculations, and modeling. Tables present specific data points or values, providing a visual representation of how variables are related or how they change.
Variables:
Equations involve variables, which can be manipulated to explore different scenarios and analyze the relationship between quantities. Tables display specific values or data points for variables, showing how they vary or are related based on the given information.
Visualization:
Equations can express relationships symbolically and provide a mathematical understanding, but they may not offer a visual representation of data. Tables provide a visual representation of data points, allowing for a quick overview and easy comparison of values.
Flexibility:
Equations offer flexibility in terms of adjusting variables and parameters to explore different scenarios and make predictions. Tables can be easily modified or expanded by adding or removing data points to reflect new observations or additional information.
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Nancy purchased from bed sheets for $36 they were on sale for 25% off what was the original price of the bed sheets
Answers
Answer:
ans $48
Step-by-step explanation:
36$ with 25%off
so
$36*75%=48
given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region?
Answers
The percentage of the time will its value falls in the indicated region are (a) 68.54%, (b) 95.44%, (c) 95.99%, (d) 99.48%, (e) 5.54%, (f) 97.19%, and (g) 15.96%.
(a) The percentage of time that a variable with 10 degrees of freedom will fall between -1.81 and 1.81 is 68.54%.
(b) The percentage of the time that a variable with 10 degrees of freedom will fall between -3.17 and 3.17 is 95.44%.
(c) The percentage of the time that a variable with 24 degrees of freedom will fall between -2.06 and 2.06 is 95.99%.
(d) The percentage of the time that a variable with 24 degrees of freedom will fall between -3.75 and 3.75 is 99.48%.
(e) The percentage of the time that a variable with 23 degrees of freedom will fall outside the interval from -2.81 to 2.81 is 5.54%.
(f) The percentage of the time that a variable with 24 degrees of freedom will fall to the right of 2.80 is 97.19%.
(g) The percentage of the time that a variable with 10 degrees of freedom will fall to the left of -1.81 is 15.96%.
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--The given question is incomplete; the complete question is
"Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? (Round your answers to one decimal place.)
(a) 10 df, between -1.81 and 1.81
(b) 10 df, between -3.17 and 3.17
(c) 24 df, between -2.06 and 2.06
(d) 24 df, between -3.75 and 3.75
(e) 23 df, outside the interval from -2.81 to 2.81
(f) 24 df, to the right of 2.80
(g) 10 df, to the left of -1.81"--
Question 11
Erica recorded the number of sales she made during each of the first 4 months when she began her new career. Erica plotted the data on a coordinate grid, and she noted that the data correlate to an exponential equation, as shown What is the domain of the function shown in the graph?
Answers
The domain of the graph is {1, 2, 3, 4}
What is Domain?"All the values" that are input into a function are referred to as the domain of a function. The set of all potential inputs for a function is its domain.
Given:
As, we know domain of a function is the set of all possible inputs for the function.
from the Graph, we can find the coordinates ( 1, 10), (2, 20), (3, 40) and ( 4, 80)
So, the domain of the graph is {1, 2, 3, 4}
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A square is shown below. The length of the diagonal can be determined by using the Pythagorean theorem. Find the rational length of one of the sides that result in a rational hypotenuse. Also find an irrational length of one of the sides that result rational hypotenuse.
Answers
step 1
Find out the hypotenuse
Applying the Pythagorean Theorem
Let
H -----> hypotenuse
\(\begin{gathered} H^2=s^2+s^2 \\ H^2=2s^2 \\ H=s\sqrt{2} \end{gathered}\)Part b
find an irrational length of one of the sides that result in the rational hypotenuse
For s=√2 -----> is an irrational number
but
H=√2*√2=2 -----> is a rational number
Part a
Find the rational length of one of the sides that result in a rational hypotenuse
Remember that
The product of a rational number and an irrational number is always irrational.
therefore
there is no rational number for one of the sides that result in a rational hypotenuseWhich angle in the triangle with vertices A (4, 4), B (5,6), and C (8,2) is a right angle?
Answers
The triangle does not represent aright-angled triangle
The coordinates of the triangle are given as:
\(\mathbf{A = (4,4)}\)
\(\mathbf{B = (5,6)}\)
\(\mathbf{C = (8,2)}\)
Start by calculating the slopes of the lines that pass through the vertices of the triangles using:
\(\mathbf{m = \frac{y_2 - y_1}{x_ 2 -x_1}}\)
So, we have:
\(\mathbf{m_{AB} = \frac{6- 4}{5 -4}}\)
\(\mathbf{m_{AB} = \frac{2}{1}}\)
\(\mathbf{m_{AB} = 2}\) --- the slope of line AB
\(\mathbf{m_{CB} = \frac{6- 2}{5 -8}}\)
\(\mathbf{m_{CB} = \frac{4}{-3}}\)
\(\mathbf{m_{CB} = -\frac{4}{3}}\) --- the slope of line CB
\(\mathbf{m_{AC} = \frac{4- 2}{4 -8}}\)
\(\mathbf{m_{AC} = \frac{- 2}{-4}}\)
\(\mathbf{m_{AC} = \frac{1}{2}}\) --- the slope of line AC
None of the slopes have the following relationship with another slope
\(\mathbf{m_1 = -\frac{1}{m_2}}\) --- condition for perpendicular line
Hence, the triangle is not a right-angled triangle
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You just obtained a credit card. You immediately purchase a stereo system for $200. your credit limit is $1000. Let's assume that you make no payments and purchase nothing more and there are no other fees. The monthly interest rate is 1.42%.
What is the growth rate of your credit card balance?
A. 0.42
B. 0.0142
C. 14.2
D. 142
Answers
Answer:
its b
Step-by-step explanation:
have a great day <3
The growth rate of your credit card balance is approximately 0.0142.
Therefore, the correct option is B.
Given that you got a credit card, you purchased a stereo system for $200. your credit limit is $1000.
The monthly interest rate is 1.42%.
We need to determine the growth rate of your credit card balance,
To calculate the growth rate of your credit card balance, we need to consider the monthly interest rate and the initial purchase amount.
The monthly interest rate is 1.42%, which can be expressed as a decimal by dividing it by 100: 0.0142.
The initial purchase amount is $200.
Assuming no payments are made and no additional purchases are made, the credit card balance will increase each month due to the accrued interest.
To calculate the growth rate, we can divide the accrued interest by the initial purchase amount.
Accrued interest = Monthly interest rate x Initial purchase amount
= 0.0142 x $200
= $2.84
Growth rate = Accrued interest / Initial purchase amount
= $2.84 / $200
≈ 0.0142
Therefore, the growth rate of your credit card balance is approximately 0.0142, which corresponds to option B.
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what is 8 1/2 times 11
Answers
Answer:
93 1/2
Step-by-step explanation:
tired of being single, but how do u do this problem?
Answers
Answer:
ok that is so hard I CAN not do it
Answer:
i am tired of being single but i do not no at all that seems really hard i feel bad for u
Step-by-step explanation:
Evaluate the expression "12 - (-36) / -6"
Answers
Answer:
ah bbhgggvgujjbgg
hhhjj
Help pleaseeee thanks
Answers
Answer:
I'd say A, Adjacent
Step-by-step explanation:
I think its A because adjacent means together/along with, and these angles are next to each other.
Can you please help I don’t understand
Answers
Answer:
( -4 , -4 )
Step-by-step explanation:
Match each function formula with the corresponding transformation of the parent function y= -4x
Answers
The transformations to the parent function y = x to obtain the function y = -4x are given as follows:
Vertical stretch by a factor of 4.Reflection over the x-axis.How to define the transformations?The functions for this problem are given as follows:
Parent function is y = x.Transformed function is y = -4x.When a function is multiplied by 4, we have that it is vertically stretched by a factor of 4.
As the function is multiplied by a negative number, we have that it was reflected over the x-axis.
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