The absolute maximum and minimum values of the function f(x) = ln(x^2 + 5x + 12) on the interval [-3, 1] are f(1) = ln(18) (maximum) and f(-2) = ln(4) (minimum).
Explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = ln(x2 + 5x + 12) on the interval [-3, 1], we first need to find the critical points of the function within this range. The critical points are those at which the derivative equals zero or does not exist.
The derivative of this function is f'(x) = 2x + 5 / (x2 + 5x + 12). Setting this equal to zero and solving for x, we get the values -1 and -2. Both of these are within our interval, so they are critical points.Next, we evaluate the function at the critical points and at the end points of the given interval. We get f(-3) = ln(6), f(-2) = ln(4), f(-1) = ln(6), and f(1) = ln(18).The absolute maximum value within the interval is ln(18) and the absolute minimum value is ln(4).
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The absolute maximum value of f(X) on the interval [-3, 1] is approximately ( 2.890), which occurs at ( x = 1), and the absolute minimum value is approximately (1.749 ), which occurs at ( x = -5/2).
To find the absolute maximum and minimum values of[tex]\( f(x) = \ln(x^2 + 5x + 12) \)[/tex] on the interval [tex]\([-3, 1]\)[/tex], we first need to find the critical points and check the values of the function at those points and at the endpoints of the interval.
1. Find the critical points: Critical points occur where the derivative of the function is zero or undefined.
First, let's find the derivative of (f(x):
[tex]\[ f'(x) = \frac{1}{x^2 + 5x + 12} \cdot (2x + 5) \][/tex]
Setting ( f'(x)) equal to zero:
2x + 5 = 0
x = -5/2
Since this value x = -5/2 lies within the interval [-3, 1], it's a critical point.
2. Check the endpoints and critical point:
[tex]- \( f(-3) = \ln((-3)^2 + 5(-3) + 12) = \ln(9 - 15 + 12) = \ln(6) \)[/tex]
[tex]- \( f(1) = \ln(1^2 + 5(1) + 12) = \ln(1 + 5 + 12) = \ln(18) \)[/tex]
[tex]- \( f\left(-\frac{5}{2}\right) = \ln\left(\left(-\frac{5}{2}\right)^2 + 5\left(-\frac{5}{2}\right) + 12\right) = \ln\left(\frac{25}{4} - \frac{25}{2} + 12\right) = \ln\left(\frac{25 - 50 + 48}{4}\right) = \ln\left(\frac{23}{4}\right) \)[/tex]
3. Compare the values:
- ln(6) = 1.792
- ln(18) = 2.890
- [tex]ln\left(\frac{23}{4}\right)[/tex] = 1.749
So, the absolute maximum value of f(X) on the interval [-3, 1] is approximately ( 2.890), which occurs at ( x = 1), and the absolute minimum value is approximately (1.749), which occurs at ( x = -5/2).
I need help on this question
Where does the graph of the function y=tan(x)
have asymptotes?
The graph of the function y=tan(x) has vertical asymptotes at odd multiples of pi/2 and periodic asymptotes at even multiples of pi.
The graph of the function y=tan(x) has asymptotes at x-values where the tangent function is undefined. The tangent function is undefined at x-values that are odd multiples of pi/2, such as -pi/2, pi/2, -3pi/2, etc. These x-values are called vertical asymptotes. The graph also has periodic asymptotes at x-values that are even multiples of pi, such as 0, pi, -pi, 2pi, etc. These x-values are called horizontal asymptotes.
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Write 760% as a decimal and as a simplified fraction
7.6
760/100
76/10
7 6/10
760% can be expressed as a decimal by simply dividing by 100, resulting in 7.6. The same percent can be expressed as a simplified fraction by writing it over 100 and simplifying, resulting in 38/5.
Explanation:To convert 760% into a decimal and a simplified fraction, we follow the process of converting percents. First, to convert a percent to a decimal, we need to divide the percent by 100. So 760% becomes 7.6 when expressed as a decimal.
Then, to convert the percent to a simplified fraction, we write the percent as a fraction over 100. So, 760% would be written as 760/100. But it can be simplified. Both 760 and 100 can be divided by 20, so when simplified the fraction becomes 38/5.
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Which expression is equivalent to 24 - (-6), minus, left parenthesis, minus, 6, right parenthesis?
Find the critical value za/2 that corresponds to a 82% confidence level.
The critical value (or z-score) that corresponds to a 82% confidence level is approximately ±1.34, which is determined using a standard normal distribution table related to the value calculated from α/2 = 0.09.
Explanation:To find the critical value za/2 that corresponds to a 82% confidence level, we start by knowing that confidence level (CL) is the area in the middle of the standard normal distribution. We know that CL = 1 - α, hence for this particular question, α = 1 - 0.82 = 0.18, and α/2 = 0.09. The Z score associated with α/2 = 0.09 in the standard normal distribution table, which represents the critical value za/2, is approximately ±1.34.
So for an 82% confidence level, the critical value za/2 is roughly ±1.34.
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HELP ASAP!!!!!!
Solve and graph the inequalities below.
Describe your graphs using words as illustrated in the sample below.
SAMPLE: Part 1: The solution is x > 5 and x<=9.
Part 2: There would be an open circle on 5 and a closed circle on 9.
Part 3: The shading is between the two numbers.
Be sure that you have included all three parts when answering each question. "Greater than or equal to" would be typed >=. "Less than or equal to" would be typed <=.
1.) d - 1 > -3 and d + 2 < 5
2.) -3k + 7 > -5 and -7k + 5 < -2
3.)-5 < 2y - 3 < 23 (Hint: refer to example 2 in the lesson.)
4.)x - 4 < 5 or x + 2 > 15
5.) 5x - 4 > 6 or -4x + 5 > 1
Which set of coordinates, when paired with (3, −3) and (10, −3), result in a square? (−3, −3) and (−10, −3) (3, 10) and (10, 3) (−3, 4) and (−10 , 4) (3, 4) and (10, 4)
Cody is twice as old as Evan and seven years ago the sum of their age was 7. find the age of each boy now
Cody is currently 14 years old and Evan is 7 years old, as determined by setting up an equation based on the information provided and solving for their ages.
Let's set up an equation to solve this problem by letting the current age of Evan be x. Since Cody is twice as old as Evan, Cody's current age would be
2x. We are told that seven years ago, their ages added up to 7. Therefore, we can express this as (x - 7) + (2x - 7) = 7. Simplifying this equation gives us 3x - 14 = 7. Adding 14 to both sides gives us 3x = 21. Dividing both sides by 3 gives us x = 7, which is Evan's current age. Therefore, Cody's age is 2x which is 2*7 = 14. Cody is 14 years old and Evan is 7 years old now.
Q #4
[tex] < 6 < 1 < 2 < 4[/tex]
Every 6 months Reuben lopez puts 420$ into an account paying 10% compounded semiannually. Find the account balance after 15 years.
The above question is related to the future value of an amount saved at regular intervals.
We need to find the future value of the $ 420 saved every six month which are compounded semi-annually.
When the compounding is semi-annually, the same is divided by 2 and the time is multiplied by 2.
The future value will be calculated as -
Future value = $ 420 × FVIFA for 30 years @ 5 %
Here, the annual rate was 10 % but since compounding is semi-annually, thus it is divided by 2. Similarly, time is multiplied by 2 i.e 15 years X 2 = 30 periods
Referring the FVIFA table, we got -
FVIFA for 30 years @ 5 % = 66.4388
Future value = $ 420 × 66.4388 = $ 27,904.30
Your principal surprises you by buying your class a turtle. He brings the turtle to your class along with a sheet from the pet store titled "Turtle Tank Rule."
*Tank walls must be at least 1 foot tall so the turtle can't climb out.
*There must be at least 400 square inches of floor space for the turtle to walk around on.
Your teacher says the volume of the tank must be smaller than 5000 cubic inches so it doesn't take up too much room in the classroom.
Give the dimensions of a tank that would work for your new turtle. Use words and numbers to explain how your tank satisfies the "Turtle Tank Rules" and your teacher requirements.
Volume of a rectangular prism= length X width X height
To satisfy the 'Turtle Tank Rules' and your teacher's requirements, the tank dimensions must meet certain criteria. You can use the formula for the volume of a rectangular prism to find the dimensions.
Explanation:To satisfy the 'Turtle Tank Rules' and your teacher's requirements, the tank dimensions must meet the following criteria:
The tank walls must be at least 1 foot tall.The tank floor must have a minimum of 400 square inches of space.The volume of the tank must be smaller than 5000 cubic inches.Let's assume the tank has length, width, and height as its dimensions. By using the formula for the volume of a rectangular prism (Volume = length * width * height), we can manipulate it to find the dimensions:
Choose a reasonable value for the height (for example, 6 inches).Calculate the area of the tank floor by dividing 400 square inches by the chosen height.Find the length and width that multiply to give this area while satisfying the volume requirement of less than 5000 cubic inches.A printer can print 27 pages in 4.5 minutes. How much time does it need to print 324 pages?
Take a look at the figure. What is the value of y? A. y = 16 B. y = 64 C. y = 14 D. y = 32
What is the solution to the expression –2 – 3 – (–4)?
The Volume of the cone shown is 6 cubic inches. What is the height of a cone with the same base diameter but a volume of only 3 cubic inches?
Answer:
3 cubic inches
Step-by-step explanation:
Using the formula, volume of the cone.
Subst. the known value of into the equation.
Thus, the volume of cone= 1(3)= 3 cubic inches.
Jeff put a small box that is 12 inches long, 8inches wide, and 4 inches tall inside a box that is 20 inches long, 15 inches wide and 9 inches high. How much space is left in the larger box?
Given the following table, determine the missing value. Growth Factor 1.13 ? 1.28 1.97 Percent 13% 256% 28% 97%
a.
256
c.
3.56
b.
2.56
d.
35.6
Answer:
missing value in the table is 3.56
Step-by-step explanation:
Percent 13% 256% 28% 97%
Growth Factor 1.13 ? 1.28 1.97
To find growth factor first we remove the percentage and then we add 1
we are given with 246 %
to remove percentage we divide by 100
[tex]\frac{256}{100} = 2.56[/tex]
To get growth factor we add 1
1+2.56= 3.56
so growth factor is 3.56
missing value in the table is 3.56
Devon wants to know if drinking milk before bed helps teenagers sleep. he chooses 10 friends on his high school basketball team for his study. every night for one month the friends drink a glass of milk before bed and later record how many hours they sleep that night. Describe three reasons why Devon's study design is flawed.
Answer:
The three reasons Devon's study is flawed because:
1. The sample size is very small. You cannot conclude anything based on small samples.
2. Sample is biased as Devon chose basket ball players and such players have proper sleep pattern and waking up pattern as compared to normal people. So, the study cannot be generalized.
3. Devon must have studied their sleep pattern before experimenting also, so that some quantitative data on change is available for conclusion.
Which expression is equivalent to -6(3x-2/3)?
A) -2x + 9
B) -3x - 6 2/3
C) -18x + 12
D) -18x + 4
Answer:
D hope this helps have a wonderful safe day
Step-by-step explanation:
Which of the following is a statistical question?
A.
What is Monica's brother's age?
B.
How many students are attending Jessica's dance class today?
C.
How many pages are there in this book?
D.
How many cars are parked in the parking lot each day?
Statistics is the discipline or science of gathering and analyzing vast amounts of numerical data. The correct option is D.
What are statistics?Statistics is the discipline or science of gathering and analyzing vast amounts of numerical data, particularly with the aim of inferring proportions in a whole from those in a representative sample.
In statistics, a large amount of data is collected to gather some sort of information, through which the important decisions can be taken. Therefore, the question that is an example of a statistical question is "How many cars are parked in the parking lot each day?".
Hence, the correct option is D.
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hey can you please help me posted picture of question
find the area of the regular decagon if the apothem is 2.8ft and a side is 1.8ft
hey can you please help me posted picture of question
Find the length of the arc shown in red. Leave your answer in terms of Pi
The calculated length of the red arc is 4.71 units
Finding the length of arc DF
From the question, we have the following parameters that can be used in our computation:
Central angle = 30 degrees
Radius = 9 meters
Using the above as a guide, we have the following:
Arc length = central angle/180 * 3.14 * Radius
Substitute the known values in the above equation, so, we have the following representation
Arc length = 30/180 * 3.14 * 9
Evaluate
Arc length = 4.71
Hence, the length of the arc is 4.71 meters
A complex fraction is a fraction of a fraction.
True
False
Answer:
The answer is TRUE
Step-by-step explanation:
Credit card A offers an APR of 27.29%, compounded monthly, while credit card B offers an APR of 27.12%, compounded daily. All else being equal, which card offers the better deal for the consumer?
The effective interest rate, also known as the effective annual interest rate, annual equivalent rate, or simply effective rate. The card which offers a better deal to the customer is Credit card A.
What is an effective rate of interest?
The effective interest rate, also known as the effective annual interest rate, annual equivalent rate, or simply effective rate, is the interest rate on a loan or financial product that is restated from the nominal interest rate and expressed as the equivalent interest rate if compound interest was payable annually in arrears. It is calculated by the formula,
EIR = [1+(i/n)]ⁿ - 1
The effective rate of interest on Credit Card A is,
Effective rate = [1+(i/n)]ⁿ - 1 = [1+(0.2729/12)]¹² - 1 = 30.97%
The effective rate of interest on Credit Card B is,
Effective rate = [1+(i/n)]ⁿ - 1 = [1+(0.2712/365)]³⁶⁵ - 1 = 31.14%
Since the effective rate of Credit card A is less than credit card B, therefore, the card which offers a better deal to the customer is Credit card A.
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A single fair die is tossed. find the probability of rolling a number greater than 55.
Find the amount of the discount on a $234 item with a discount of 15%. A. $40.00 B. $23.40 C. $35.10 D. $35.01
Ethan made 14 out of 25 foul shots at basketball practice what percent of shots did he make
Someone help I don't understand this.