y = f(x) has the derivative f'(x) = (x + 1)²(x + 3)(x² + 2mx + 5) with ∀x∈i.
find m so that g(x) = f(|x|) has exactly 1 extreme point
Answers
9514 1404 393
Answer:
m ≥ -√5
Step-by-step explanation:
If g(x) = f(|x|) for x∈i, then g(x) = f(x) for x∈ℝ: x ≥ 0.
f'(x) is 5th-degree, so f(x) is 6th-degree, meaning it is generally U-shaped. Since we're only concerned with x ≥ 0, we want to make sure f'(x) has no real zeros of odd multiplicity such that x > 0. The given factors of f'(x) make it have real zeros at x = -3 and x = -1.
For the last factor, (x² +2mx +5) to have no positive real zeros of odd multiplicity, we must have m ≥ 0 or the discriminant ≤ 0. The discriminant is ...
d = b² -4ac = (2m)² -4(1)(5) = 4m² -20 . . . . . discriminant of the last factor
d ≤ 0 . . . . . . . . . . the condition for no real zeros
4m² -20 ≤ 0
m² -5 ≤ 0 . . . . . . divide by 4
m² ≤ 5 . . . . . . . . .add 5
|m| ≤ √5 . . . . . . . take the square root
This tells us there will be a positive real zero of multiplicity 2 in f'(x) when m = -√5, and there will be no positive real zeros for -√5 < m < 0
There will be no odd-multiplicity positive real zeros in the derivative function f'(x) as long as m ≥ -√5. This means the slope of f(x) is non-negative for x ≥ 0, hence f(|x|) has its only minimum at x=0.
_____
Additional comment
The multiplicity of the zeros of f'(x) is important because the derivative will only change sign where the multiplicity is odd. When the discriminant of (x²+2mx+5) is zero, the associated positive real zero will have multiplicity 2, hence f'(x) will not change sign there.
Related Questions
Lois is 3 years older than Oscar. The sum of their ages is no more than 20. Write an inequality that can be used to represent this situation, where x represents Oscar’s age.
Answers
ATQ
[tex]\\ \sf\longmapsto x+x+3\leqslant 20[/tex]
[tex]\\ \sf\longmapsto 2x+3\leqslant 20[/tex]
[tex]\\ \sf\longmapsto 2x\leqslant=17[/tex]
[tex]\\ \sf\longmapsto x\leqslant \dfrac{17}{2}[/tex]
Answer:
let Oscar's years be x
• Oscar → x
• Lois → (x + 3)
[tex]{ \rm{x + (x + 3) \leqslant 20}} \\ \\ { \boxed{ \rm{2x + 3 \leqslant 20}} }[/tex]
Find the average speed in km/h for a journey of 175 meters and 35 seconds
Answers
Answer:
18 km/h
Step-by-step explanation:
1. Convert 175 metres into km
1000 m = 1 km (divide by 1000)
175 ÷ 1000 = 0.175 km
so 175 m = 0.175 km
2. Convert 35 seconds into hours
60 seconds = 1 min
60 mins = 1 hour
3600 (60 × 60) seconds = 1 hour (divide by 3600)
35 ÷ 3600 = 0.00972...
3. Calculate the speed
speed = distance ÷ time
= 0.175 ÷ 0.00972...
= 18 km/h
Go to your math tools and open the Graph tool to graph the equation you found for calculating profit. Set the scale of the graph to go from -5 to 40 on the x-axis and from -80 to 80 on the y-axis.
Part A
What do the points on the line in quadrant IV represent in terms of the situation?
Answers
Answer:
The y-coordinates in quadrant IV are negative. Because y represents profit, negative values of y signify not making a profit. That is, negative values signify loss. The points in quadrant IV represent the baseball team losing money.
Step-by-step explanation:
I took the assignment :).
Ann is helping her aunt sew a lace border onto 2 quilts. Each quilt is 60 inches wide by 80 inches long. At the craft store, they buy a 25-yard roll of lace. How many inches of lace will they have left after finishing the quilts?
Answers
Answer:
340 inches
Step-by-step explanation:
25 yds = 900 in
60 + 80 + 60 + 80 = 280
280 x 2 = 560
900 - 560 = 340
please someone help me giving brainliest :)
Answers
6 goes into 24 4 times and since both the top and bottom are negative the answer is positive
Answer:
It should be 4.
Question 10 of 2
What is the coefficient in the expression 8x2 +1?
O A. 1
ОВ. x
O C. 2
O D. 8
What is the coefficient?
Answers
Therefore, the correct answer is D) 8.
Please mark my answer as the Brainliest if you find this helpful :)
Solve the simultaneous equations 2x + y = 5 and 2y = x – 10.
Answers
The first equation can be substituted into the second one. It would be: 2(-2x+5)=x-10 and after solving it, x = 4
[tex]▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ { \huge \mathfrak{Answer}}▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ [/tex]
The given equations are :
[tex]2x + y = 5[/tex][tex]2y = x - 10[/tex]From the first equation, we get :
[tex]2x + y = 5[/tex][tex]y = 5 - 2x[/tex]now, let's put the value of y as (5 - 2x) in second equation to find the value of x :
[tex]2y = x - 10[/tex][tex]2(5 - 2x) = x - 10[/tex][tex]10 - 4x = x - 10[/tex][tex]5x = 20[/tex][tex]x = 4[/tex]hence, value of x = 4, Plugging the value of x in second equation :
[tex]2y = x - 10[/tex][tex]2y = 4 - 10[/tex][tex]2y = - 6[/tex][tex]y = - 3[/tex]so, the required values are :
x = 4 y = -3Find the x- and y-intercepts of : 4y-6x = 24
Answers
Answer:
x-intercept: (-4, 0)
y-intercept: (0, 6)
Step-by-step explanation:
4y - 6x = 24
To find the x-intercept, set the y value equal to 0.
4(0) - 6x = 24
Simplify by removing 4(0) since it just equals 0.
-6x = 24
Divide both sides of the equation by -6
x = -4
Since we set y equal to 0 and we know that the x value is -4, we can create a coordinate for the x-intercept of the equation.
(x, y) = (-4, 0)
Repeat the process for finding the y-intercept, but set x equal to 0.
4y - 6(0) = 24
4y - 0 = 24
4y = 24
Divide both sides of the equation by 4
y = 6
Write as a coordinate/intercept.
(x, y) = (0, 6)
Let me know if you have any questions!
If (x+2,5)=(3y+2)find x and y
Answers
Answer:
X = 7
Step-by-step explanation:
because of you do 5+2 = 7 then you minus 3 which you get 4.
the. if you have a production of around 10 you will get 7.
or you can do.
3+2 which is 5 and then you multiply that by 2 which is 10 and then you comprised that number therefore you will get 6 but you have to find X so we youndelie (think that's how u spell it) X which will equal to 1 and 6+1 is 7. once you get it it's quite simple
Original grade:80
New grade:100
Find the precent of increase
Answers
Answer:
25%
Step-by-step explanation:
because from 80 to 100 is 20 and 20 is 1/4 of 80 and 1/4 to a decimal is .25
Answer:
80/100. That's 20% increase. When you hadd 8 and 2 you get 10.
Step-by-step explanation:
A gym employee earns the same amount each month. After working for three months, he earned $4,500. Complete the table to determine how much money he will make over a five-month period. 5.Is the relationship proportional? Explain your thinking.
Answers
Answer:
1,500
Step-by-step explanation
1. to divide 3 by 4,500.
after dividing 3 and 4,500 you get
1,500
The money he will make over a five-month period will be $7,500.
What are ratio and proportion?A ratio is a group of sequentially ordered numbers a and b expressed as a/b, where b is never equal to zero. When two objects are equal, a statement is said to be proportional.
A rec center worker procures a similar sum every month. In the wake of laboring for a considerable length of time, he procured $4,500.
The amount of money that he earns each month is given as,
⇒ $4,500 / 3
⇒ $1,500 per month
Then the money he will make over a five-month period will be given as,
⇒ $1,500 x 5
⇒ $7,500
The money he will make over a five-month period will be $7,500.
More about the ratio and the proportion link is given below.
https://brainly.com/question/14335762
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I need help with #7 asap plss
Answers
Answer:
sorry i would help if i new let me see if i can find it somewhere
Step-by-step explanation:
please help me , thank you
Answers
Hello BinibiningDel!
[tex] \huge \boxed{\mathbb{QUESTION} \downarrow}[/tex]
Factorise the polynomials.[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
[tex] \tt \: 1) \: \: \: {x}^{2} - 49 \\ \tt \: = (x) ^{2} - ( 7) ^{2} \\ = \boxed{ \bf \: (x + 7)(x - 7) }[/tex]
Method used :-
Use the algebraic identity ⇨ a² - b² = (a + b) (a - b ).Here, a = x & b = 7.__________________
[tex] \tt 2) \: \: \: {a}^{2} + 10a+ 25 \\ = \boxed{\bf(a + 5)^{2} }[/tex]
Method used :-
Use the algebraic identity ⇨ a² + 2ab + b² = (a + b)².Here, a = a & b = 5.__________________
[tex] \tt \:3) \: \: \: {x}^{2} + x - 30 \\ = \tt ({x}^{2} - 5x) + (6x - 30) \\ = \tt \: x(x - 5) + 6(x - 5) \\ = \boxed{ \bf(x + 6)(x - 5)}[/tex]
Method used :-
Use the splitting the middle term method to factorise this.__________________
[tex] \tt \: 4) \: \: \: {4x}^{2} - {y}^{2} \\ = \tt \: (2x) ^{2} - (y) ^{2} \\ = \boxed{ \bf(2x - y)(2x + y)}[/tex]
Method used :-
Use the algebraic identity ⇨ a² - b² = (a + b) (a - b ).Here, a = 2x & b = y.__________________
[tex] \tt \: 5) \: \: \: 8 {m}^{2} n ^{3} - 6mn ^{2} \\ = \tt \: 2\left(4m^{2}n^{3}-3mn\right) \\ = \boxed{ \bf \: 2mn\left(4mn^{2}-3\right) }[/tex]
Method used :-
Take the common factor out & factorise it.__________________
[tex] \tt \: 6) \: \: \: \tt \: 2 d ^ { 2 } + 5 d - 12 \\ = \tt \: \left(2d^{2}-3d\right)+\left(8d-12\right) \\ \tt \: = d\left(2d-3\right)+4\left(2d-3\right) \\ = \boxed{ \bf \: \left(2d-3\right)\left(d+4\right) }[/tex]
Method used :-
Use the splitting the middle term method to factorise this.__________________
[tex] \tt \:7) \: \: \: y ^ { 3 } + 27 \\ = \tt (y) ^{3} + ( {3})^{3} \\ = \boxed{ \bf\left(y+3\right)\left(y^{2}-3y+9\right) }[/tex]
Method used :-
Use the algebraic identity ⇨ a³ + b³ = (a + b) (a² - ab + b²).Here, a = y & b = 3.__________________
[tex] \tt \: 8) \: \: \: 20 b ^ { 3 } c ^ { 5 } + 35 b ^ { 4 } c ^ { 2 } \\ = \tt \: 5\left(4b^{3}c^{5}+7b^{4}c^{2}\right) \\ = \boxed{ \bf \: 5b^{3}c^{2}\left(4c^{3}+7b\right) }[/tex]
Method used :-
Take the common factor out & factorise it.__________________
Hope it'll help you!
ℓu¢αzz ッ
Please help me solve this ASAP
Answers
Answer:
x = 1.6Step-by-step explanation:
[tex] - 2.5(4x - 4) = - 6 \\ - 2.5(4x) - 2.5( - 4) = - 6 \\ - 10x + 10 = - 6 \\ - 10x + 10 - 10 = - 6 - 10 \\ - 10x = - 16 \\ \frac{ - 10x}{ - 10} = \frac{ - 16}{ - 10} \\ x = 1.6 \\ [/tex]
An amount of 3560 is decreased by 35.5% every year how much will it be worth after eight years
Answers
3,560 divided by 35.5 = 101.7
101.7 • 8 = 813.6
3,560 - 813.6 = 2,746.4
which of the following represents a linear function with slope of 6?
Answers
Answer:
any equation with a 6 before the x
y=6x+b
Step-by-step explanation:
standard form is y=mx+b
m is the slope and b is y-intercept
What number is 5.1% of 15?
Answers
Answer: 0.765
Step-by-step explanation:
“of” means “multiply”
5.1% can be written as 0.051
Multiply 15 by 0.051
0.765
I need help with this one
Answers
Answer:
S = 180C²
18,000 lbs
Step-by-step explanation:
Assuming the table attached with C in inches and S in pounds, we see that the first differences in S are ...
180, 540, 900
These are not constant, so the relation is not linear.
Second differences are ...
360, 360
These are constant, so the relation is quadratic. With a short trial, we find that a suitable equation is ...
S = 180·C²
__
The safe load for C=10 is then ...
S = 180·10² = 18,000 . . . pounds
better?
Mary has saved $20.50 in the past 3 weeks. At this rate, how much will she save in 18 weeks?
Answers
Answer: $123
Step-by-step explanation:
$20.50/3 (weeks) = 6.83333333333
$6.83333333333 x 18 (weeks) = $123
Answer:
Step-by-step explanation:
18 weeks = 6 * 3
= 6 * 20.5
= $ 123
what is 6+6=blank find the answer please
Answers
An atom of silicon is approximately 0.00000000025 meters wide. How do you express that number in scientific notation? O 25 10-10 meters O 2.5 × 10-11 meters O 2.5 1010 meters O 2.5 x 10-10 meters
Answers
Answer:
.00000000025 = 2.5 * 10^-10
Looks like the last one
Can also write
2.5E-10
please help this is a graded assignment due tomorrow
Answers
b° is 160°
c° is 15°
d° is 75°
show steps and i'll give brainly
Answers
Answer:
FOIL F-FIRST O-OUTSIDE I-INSIDE L-LAST
Step-by-step explanation:
[tex]2.6 = \times.2 \frac2.{68}{?} [/tex]
Describe the end behavior of y = 5x^4
Answers
Answer:
answer below
Step-by-step explanation:
5x⁴: degree even (4), leading coefficient +5, rise to left and rise to right
x -> -∞ y -> ∞
x -> ∞ y -> ∞
Which of the following best compares the slopes and y-intercepts of the two functions? (1 point)
The slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).
The slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is greater than the y-intercept of g(x).
The slope of f(x) is equal to the slope of g(x). The y-intercept of f(x) is less than the y-intercept of g(x).
The slope of f(x) is equal to the slope of g(x). The y-intercept of f(x) is greater than the y-intercept of g(x).
PLS HELP
Answers
Answer:
The slope of f(x) is equal to the slope of g(x). The y-intercept of f(x) is greater than the y-intercept of g(x).
Step-by-step explanation:
F(x) is already given to you on the graph. You need to plot the x and g(x) graph. (remember g(x) is just a fancy way of saying y)
Once this line is plotted, you will see that f(x) and g(x) are parallel to one another. G(x) is below f(x), so its y-intercept is lower.
Find the missing amount)
Original: $160
New: $136
%Decrease:_______
(not multiple choice)
Answers
Cross multiply and divide:
136*100=13600
13600/160=85
Answer: 85%
3/5e-6=-2/5(e-10)-7
Answers
Answer:e=19
Step-by-step explanation:
3/5e-6 = -2/5e +20 -7 distributed multiplication into brackets
3/5e + 2/5e = 20-7+6 collected like terms
e=19
Which one of the following would most likely have a positive linear correlation coefficient?
A. grade point average compared to the number of hours a student watches TV
B. the height of a tomato plant throughout the year
C. a person's salary compared to the amount of education they have attempted
D. SAT scores compared to hours spent studying
Answers
Answer: B
Step-by-step explanation: The tomato grows more over time so it will be a positive..
The height of a tomato plant throughout the year shows a positive linear correlation coefficient option (B) is correct.
What is correlation?It is defined as the relation between two variables which is a quantitative type and gives an idea about the direction of these two variables.
[tex]\rm r = \dfrac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{{[n\sum x^2- (\sum x)^2]}}\sqrt{[n\sum y^2- (\sum y)^2]}}[/tex]
A. grade point average compared to the number of hours a student watches TV.
The grade points depend on the number of hours a student watches TV.
If watching time increases the grade point decreases.
B. the height of a tomato plant throughout the year
The height of a tomato and the number of years show a positive linear correlation.
C. a person's salary compared to the amount of education they have attempted:
The above situation represents an exponential regression.
D. SAT scores compared to hours spent studying
If the time spent in studying increases the SAT scores increases rapidly.
Thus, the height of a tomato plant throughout the year shows a positive linear correlation coefficient option (B) is correct.
Learn more about the correlation here:
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Patrick can paint a house by himself in
t
+
4
hours. Karen can paint the same house by herself in
t
+
9
hours. If they work together, they can paint the house in
t
hours. How much time will it take for Patrick to paint the house by himself?
Answers
Answer:
Its 12
Step-by-step explanation:
Ok done
Jake earns $8 per hour for cutting grass. If he works a 35-hour week,
how much money will he earn?
Answers
There are many ways to solve these types of questions. But I am going to use the fraction method.
Step 1: Set up a fraction.
Setting up a fraction, given the question will be the units on the same level. And the money earned will be a variable, "[tex]m[/tex]". It does not matter whether they are top or bottom.
[tex]\frac{8}{1} = \frac{m}{35}[/tex]
Step 2: Solve
[tex]\frac{8}{1} = \frac{m}{35}[/tex]
Multiply 35 to both sides[tex]\frac{35m}{35}=8\cdot \:35[/tex]
[tex]m=280[/tex]
Therefore, Jake will earn $280.