A curve has equation y=4x^3 -3x+3. Find the coordinates of the two stationary points. Determine whether each of the stationary points is a maximum or a minimum.
Answers
Answer:
There are two stationary points
Local max = (-0.5, 4)Local min = (0.5, 2)Note that 1/2 = 0.5
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How to get those answers:
Stationary points occur when the derivative is zero, when the graph is neither increasing nor decreasing (i.e. it's staying still at that snapshot in time).
Apply the derivative to get:
f(x) = 4x^3 - 3x + 3
f ' (x) = 12x^2 - 3
Then set it equal to zero and solve for x.
f ' (x) = 0
12x^2 - 3 = 0
12x^2 = 3
x^2 = 3/12
x^2 = 1/4
x = sqrt(1/4) or x = -sqrt(1/4)
x = 1/2 or x = -1/2
x = 0.5 or x = -0.5
----------------------
Let's do the first derivative test to help determine if we have local mins, local maxes, or neither.
Set up a sign chart as shown below. Note the three distinct regions A,B,C
A = numbers to the left of -0.5B = numbers between -0.5 and 0.5; excluding both endpointsC = numbers to the right of 0.5The values -0.5 and 0.5 are not in any of the three regions. They are the boundaries.
The reason why I split things into regions like this is to test each region individually. We'll plug in a representative x value into the f ' (x) function.
To start off, we'll check region A. Let's try something like x = -2
f ' (x) = 12x^2 - 3
f ' (-2) = 12(-2)^2 - 3
f ' (-2) = 45
The actual result doesn't matter. All we care about is whether if its positive or negative. In this case, f ' (x) > 0 when we're in region A. This tells us f(x) is increasing on the interval [tex]-\infty < x < -0.5[/tex]
Let's check region B. I'll try x = 0.
f ' (x) = 12x^2 - 3
f ' (0) = 12(0) - 3
f ' (0) = -3
The result is negative, so f ' (x) < 0 when [tex]-0.5 < x < 0.5[/tex]. The f(x) curve is decreasing on this interval.
The change from increasing to decreasing as we pass through x = -0.5 indicates that we have a local max here.
Plug x = -0.5 into the original function to find its paired y coordinate.
f(x) = 4x^3 - 3x + 3
f(-0.5) = 4(-0.5)^3 - 3(-0.5) + 3
f(-0.5) = 4
The point (-0.5, 4) is a stationary point. More specifically, it's a local max.
Side note: This is the same as the point (-1/2, 4) when written in fraction form.
----------------------
Let's check region C
I'll try x = 2
f ' (x) = 12x^2 - 3
f ' (2) = 12(2)^2 - 3
f ' (2) = 45
The positive outcome tells us that any number from region C does the same, and f ' (x) > 0 when [tex]0.5 < x < \infty[/tex]. The function f(x) is increasing on this interval.
Region B decreases while C increases. The change from decreasing to increasing indicates we have a local min when x = 0.5
Plug this x value into the original equation
f(x) = 4x^3 - 3x + 3
f(0.5) = 4(0.5)^3 - 3(0.5) + 3
f(0.5) = 2
The local min is located at (0.5, 2) which is the other stationary point.
The graph and sign chart are shown below.
The local min is located at (0.5, 2) which is the other stationary point.
How to Determine whether each of the stationary points is a maximum or a minimum?Stationary points occur when the derivative is zero, when the graph is neither increasing nor decreasing.
Apply the derivative to get:
f(x) = 4x^3 - 3x + 3
f ' (x) = 12x^2 - 3
Then set it equal to zero and solve for x.
f ' (x) = 0
12x^2 - 3 = 0
12x^2 = 3
x^2 = 3/12
x^2 = 1/4
x = 1/2 or x = -1/2
x = 0.5 or x = -0.5
The first derivative test to help determine if we have local mins, local maxes, or neither.
A = numbers to the left of -0.5
B = numbers between -0.5 and 0.5 excluding both endpoints
C = numbers to the right of 0.5
Thus, The values -0.5 and 0.5 are not in any of the three regions. They are the boundaries.
Let's try something like x = -2
f ' (x) = 12x^2 - 3
f ' (-2) = 12(-2)^2 - 3
f ' (-2) = 45
In this case, f ' (x) > 0 when we're in region A.
Let's check region B x = 0.
f ' (x) = 12x^2 - 3
f ' (0) = 12(0) - 3
f ' (0) = -3
The result is negative, so f ' (x) < 0 when f(x) curve is decreasing on this interval.
The change from increasing to decreasing as we pass through x = -0.5 indicates that we have a local max here.
Substitute x = -0.5 into the original function to find its paired y coordinate.
f(x) = 4x^3 - 3x + 3
f(-0.5) = 4(-0.5)^3 - 3(-0.5) + 3
f(-0.5) = 4
The point (-0.5, 4) is a stationary point, it's a local max.
Let's check region C x = 2
f ' (x) = 12x^2 - 3
f ' (2) = 12(2)^2 - 3
f ' (2) = 45
The positive outcome tells us that any number from region C does the same, and f ' (x) > 0 when The function f(x) is increasing on this interval.
Susbtitute this x value into the original equation
f(x) = 4x^3 - 3x + 3
f(0.5) = 4(0.5)^3 - 3(0.5) + 3
f(0.5) = 2
Hence, The local min is located at (0.5, 2) which is the other stationary point.
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Related Questions
Find the equation of a line passing through (-2/3, 5) that is parallel to 4x-2y=6. Express your answer in point-slope form.
Answers
Using the information given, it is found that the linear function is given by:
[tex]y - 5 = 2\left(x + \frac{2}{3}\right)[/tex]
A linear equation, in point-slope formula, is modeled by:
[tex]y - y_1 = m(x - x_1)[/tex]
In which:
m is the slope.The point is [tex](x_1, y_1)[/tex].If two lines are parallel, they have the same slope.Parallel to 4x - 2y = 6, which, in standard form is:
[tex]2y = 4x - 6[/tex]
[tex]y = \frac{4}{2}x - \frac{6}{2}[/tex]
[tex]y = 2x - 3[/tex]
Hence, the slope is [tex]m = 3[/tex].
Point [tex]\left(-\frac{2}{3}, 5\right)[/tex], hence, the equation of the line is:
[tex]y - y_1 = m(x - x_1)[/tex]
[tex]y - 5 = 2\left(x + \frac{2}{3}\right)[/tex]
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Karen wants to advertise how many chocolate chips are in each Big Chip cookie at her bakery. She randomly selects a sample of 60 cookies and finds that the number of chocolate chips per cookie in the sample has a mean of 15.9 and a standard deviation of 1.8. What is the 99% confidence interval for the number of chocolate chips per cookie for Big Chip cookies? <μ<
Enter your answers accurately to one decimal place.
Answers
Sample mean : \overline{x}=10.6x=10.6
Standard deviation : s=1.7s=1.7
Significance level : \alpha:1-0.95=0.05α:1−0.95=0.05
Critical value : z_{\alpha/2}=1.96
Hence the 95% confidence interval for the number of chocolate chips per cookie for big chip cookies= (10.1989,\ 11.0011)(10.1989, 11.0011)
Help with this problem
Answers
Answer:
The vertical intercepts is
(4/3,0) (3,0) (1,0)
The horinzontial intercepts is
(0,72)
Step-by-step explanation:
The vertical or y intercepts will be zeroes of the following factors.
3x+4=0
x-1=0
(x-3)^2=0
We solve for x each time
x=-4/3
x=1
x=3
Part B). We need to find the intercepts by setting the equation equal to zero.
The constant expanded will equal 72.
If x=0,
0y=0
where y is any real number.
So this means all the exponets will equal 0. The constant will just add to the term. so our horinzontal intercept is
Let f(x) =
=(x^2-16x+39
Divide by X-3
a,if x #3
if x = 3
Answers
Answer:
Step-by-step explanation:
x² - 16x + 39 = ( x - 3 )( x - 13 )
If f(x)= 5x, what is f^-1(x)?
Answers
Step-by-step explanation:
[tex]f( x) {}^{ - 1} \\ f(x) = 5x \\ y = 5x \\ x = 5y \\ y = \frac{x}{5} \\ f(x) {}^{ - 1} = \frac{x}{5} [/tex]
Plz HELP ME NOWWWWWWWW
Answers
Answer:
x=13
Step-by-step explanation:
(4x-20+x-1)/8 = (3x-9+4x-16)/12
(5x-21)/8=(7x-25)/12
(5x-21)/8 x 12/1 = (5x-21)/2 x 3/1 = (15x -63)/2
15x - 63 = 14x - 50
x=13
13. This shows a part of a multiplication table. Find the missing numbers. Explain how you found the numbers.
35 40
42 ?
? ?
Answers
The missing numbers in the multiplication table are 49, 48 and 56 respectively
Given:
35
42
?
35 = 7 × 5
42 = 7 × 6
? = 7 × 7 = 49
40
?
?
40 = 8 × 5
? = 8 × 6 = 48
? = 8 × 7 = 56
Multiplication table is a mathematical table showing the products of each of integers such as 7 and 5, 1 and 10, 13 and 12 etcTherefore, the missing numbers in the multiplication table are 49, 48 and 56 respectively
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11/2 ___ 55%? PLS I NEED A AWNSER
Answers
Answer:
The answer is >
Step-by-step explanation:
11/2 as a percentage is equal to 550% which is greater than (>) 55%
Write 110% as a mixed number in simplest form
Answers
Answer:
1 1/10
Step-by-step explanation:
110% = 110/100 = 11/10 = 1 1/10
What is the value of x?
Answers
again, we're assuming that both triangles are similar, and thus using proportions.
[tex]\stackrel{\textit{assuming the triangles are similar}}{\cfrac{x+1}{27}=\cfrac{5}{x-5}\implies x^2+x-5x-5=135}\implies x^2-4x-5=135 \\\\\\ x^2-4x-140=0\implies (x-14)(x+10)=0\implies x = \begin{cases} 14~\textit{\large \checkmark}\\ -10 \end{cases}[/tex]
we do not use the -10, because whatever value "x" may be, can't be negative or even 0.
PLS HELP ITS A MAJOR GRADE AND DUE TOMORROW
A rental car company charges $55 plus $0.50 per mile to rent a car. Mt. Judd does not want to spend more than
%150 for his rental car. Write and solve an inequality to find how many miles he can drive and not spend more
than $150. Interpret/explain the solution.
Incorrect work/Solution
Identify and Explain Error
m = miles
0.5m < 150
0.5 0.5
m< 30
Mr. Judd can drive at least 50 miles.
Correct Work/ Solution
Share Strategy
Answers
Answer:
*See below*
Step-by-step explanation:
Identify and Explain Error
The method shown is using fractions to compare costs. This strategy does not work due to the fact that they have not factored in the $55 he pays for the car before hand. Also, 150 divided by 0.5 does not equal 30, it equals 300 so, even if he did not pay $55 beforehand, the equation is still incorrect.
Correct Work/Solution
$55 to rent
$0.50 per mile
Let's start by removing $55 from $150 to see how many dollars is left over for gas.
150 - 55 = 95
Then, divide 95 by 0.5
95 ÷ 0.5 = 190
He can drive at least 190 miles.
Share Strategy
Since he starts off paying $55 dollars out of $150, we need to subtract $55 by $150 to see how much cash he has left over for mileage. $150 minus $55 equals $95 so, he has $95 left over for mileage. $95 will then be divided by $0.50 to find out how many miles he can drive. We are dividing by $0.50 because that's the cost per mile. $95 divided by $0.50 equals 190 so he can drive at least 190 miles.
Note:
Hope this helps :)
Have a great day!
Hue is arranging chairs. She can form 2 rows of a given length with 6 chairs left over, or 8 rows of that same length if she gets 12 more chairs. Complete the equation and solve it to find how many chairs are in that row length. In your equation, use the variable x to represent the number of chairs in each row.
Answers
Answer:
I am stuck on the same question please comment if someone answers
Answer:
3
Step-by-step explanation:
Okay so 18 chairs were used more for 6 more rows so each row has 3 length.
x = 3
Now this makes no sense, because I don't know why you would leave 6 chairs if you only have 3 chairs for each row but....(Hue is weird so let's not question him)
Anyways mathematically
first arrangement = 2x + 6
second arrangement = 8x - 12
2x + 6 = 8x - 12
6x = 18
x = 3
A store is having a clearance sale where all prices are 30% off. The store is located in a county with a 7.5% sales tax. The final price, including sales tax, of any item in the store will be what percent of the original price? Explain your reasoning.
Answers
Ans. 450
Solve.
Add 100and you will get the ans
A physician orders Fragmin 1,600 international units subcutaneously stat. There are 9-mL vials of Fragmin (dalteparin) 10,000 international units/mL available in stock. What volume of medication should be administered? a) 16 mL O b) 1.6 mL c) 0.16 mL d) 0.016 mL
Answers
Answer:
HH sheikh Zayed road Dubai UAE Tel Aviv university of California Berkeley
Which number can each term of the equation be multiplied by to eliminate the fractions before solving? -3/4m-1/2=2+1/4m
Answers
Answer:
4
Step-by-step explanation:
Looking at the fractions, -3/4 and 1/2 and 1/4, you can see the LCD or least common denominator is 4.
Thus you get -3, 2, and 1 respectively by multiplying each fraction by 4.
1. It is a way of comparing two or more quantities having the same units. * 1 point O a. decimal O b. addends . O c. ratio O d. numerals 2. Which of the following fractions is appropriate equivalent fraction of 3/4. 1 point O a. 4/8 O b. 9/12 O c. 10/15 Oc O d. 12/17
Answers
Answer:
Step-by-step explanation:
Equivalent fractions for 1/4
From this chart, we can observe that the equivalent fractions of 1/4 are: 2/8, 3/12, 4/16,... Two fractions are said to be equivalent if their values (decimal/graphical) are the same. We usually multiply the numerator and denominator of a fraction by the same number to get its equivalent fraction.
The ordered pair (8,2) is found in the ____ Quadrant
Answers
what rounds up to 5,620 to the nearest ten
Answers
5,610
Step by step explanation:When rounding to the nearest ten, like we did with 5620 above, we use the following rules:
We round the number up to the nearest ten if the last digit in the number is 5, 6, 7, 8, or 9.We round the number down to the nearest ten if the last digit in the number is 1, 2, 3, or 4.If the last digit is 0, then we do not have to do any rounding, because it is already to the ten.Answer: 5620
~~Step-by-step explanation~~
Rounding has a simple rule. 5 or above? Up you shove. 4 or below? Keep it low.
1. What are you using to round?
5620 : The tens place is the number we are either going to be rounding up or keeping. In this case, it is the 2 in 5620.5620 : When rounding to the nearest ten, you look at the ones section. In this case, it is the 0 in 5620.2. Execution
When rounding up to tens, look at the ones value. If the number is 5 or above, round the tens place up a number. (ex. 5625 --> 5630) If the number is a 4 or below, keep the tens value the same. (ex. 5624 --> 5620)Since the ones place in 5620 is below a four, we are going to keep the tens value the same, so the answer is going to be 5620.
Jorge heard that as a general rule, he should save at least 10% of his take-home pay. If Jorge’s take-home pay is $2950 per month, what is the minimum amount per year that he should save?
Answers
Answer:
$295
Step-by-step explanation:
2950 · 0.10 = 295
What is an equation of the line that passes through the points (-7, 7) and (3, 7)(
Answers
Answer:
y=7
Step-by-step explanation:
The slope is (7-7)/(3+7) = 0. So, the line is a horizontal line.
The y value of both points is 7.
The Smith family goes out for lunch, and the price of the meal is $45.The sales tax on the meal is 6%, and the family also leaves a 20%tip on the pre-tax amount. What is the total cost of the meal?
Answers
Answer:
$56.70
Step-by-step explanation:
45 x 6% = 2.7 Multiply price of meal with the percentage of sales tax
45 x 20% = 9 Multiply price of mean with the percentage of tip
$9 is the tip amount
$2.70 is the sales tax
$45 + $2.70 + $9 = $56.70. Add the price of the meal with the sales tax and tip amount to get the total cost of the meal.
Help I need the answers ASAP
Answers
Answer:
304 and 512
Step-by-step explanation:
What is the slope of the line?
Answers
Answer:
-5
Step-by-step explanation:
Use the slope triangle I mentioned in a previous question
what is 3.98.7 * 29.76+
Answers
Answer:
- 7% of 3.98...
... is equivalent to multiplying them:
- 7% × 3.98 ≈ - 0.28
(Rounded to a maximum of 2 decimals)
Step-by-step explanation:
somebody please help me:(
Answers
There are flags planted along a forest. There
are blue flags planted every 10 miles and there
are purple flags planted every 20 miles. If the
forest is 300 miles long, how many times will
there be a blue and purple flag in the same
spot?
Answers
Answer:
15 times there will be a blue and purple flag in the same spot
Step-by-step explanation:
300 miles / 10 miles = 30 blue flags in total
300 miles / 20 miles = 15 purple flags in total
There will be at least one blue flag for every purple flag. Since there is 15 purple flags in total, there will be 15 times this can occur.
I have attatched a demonstration of my thoughts to this post.
im giving my friends 9 cupcakes thats 60 percent of my order how many more cupcakes do i need to get my order to 100 percent
Answers
Answer:
6 more cupcakes.
Step-by-step explanation:
60% of 15 is 9, so 15-9 is 6.
The number of cupcakes needed to get the order to 100% is 6 cupcakes.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/100 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
10% = 10/100 = 1/10
We have,
60% = 9 cupcakes
Multiply 100/60 on both sides.
100/60 x 60% = 100/60 x 9 cupcakes.
100% = 5 x 3 cupcakes
100% = 15 cupcakes
Thus,
6 more cupcakes are needed.
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Matthew invested $3,000 into two accounts. One account paid 3% interest and the other paid 8% interest. He earned 4% interest on the total investment. How much money did he put in each account?
Answers
Answer:
600 in the 8% interest and 2400 in the 3% interest.
Step-by-step explanation:
600 + 8% = 48
2400 + 3% = 72
72 + 48 =120
If your money doubles once every 30 minutes and your start with $12, how much money will you have in 1.5 day?
Answers
Answer:
$1728
Step-by-step explanation:
1.5 days is 36 hrs
36 hrs is 2160 mins
2160/30 = 72 (this is the amount of times our money will be doubled)
72*2 = 144
$12*144 = $1728
[tex]3^{2x+1} -4.9^{x} =-9[/tex]
