If function f : R --> R where f(X)=X³ - kX² + 12X - 7 is a one to one function, then k belongs to....
Answers
Since f(x) is a cubic polynomial, it has at most 3 distinct roots. If f(x) has 3 real roots, then f(x) = 0 for more than one instance of x.
But if f(x) is one-to-one, then there must be only one real root and the other two are non-real. Let a + bi and a - bi be these non-real roots and c the single real root; then we can factorize f(x) as
[tex]f(x) = x^3 - kx^2 + 12x + 7 = (x - (a + bi)) (x - (a - bi)) (x - c)[/tex]
Expand the right side to get
[tex]f(x) = x^3 - kx^2 + 12x + 7 = (x^2 - 2ax + a^2+b^2) (x - c)[/tex]
[tex]f(x) = x^3 - kx^2 + 12x + 7 = x^3 - (2a + c) x^2 + (a^2 + 2ac + b^2) x - (a^2c + b^2c)[/tex]
from which it follows that
[tex]\begin{cases}k = 2a + c \\ 12 = a^2+2ac+b^2 \\ 7 = -a^2c-b^2c\end{cases}[/tex]
Since f(x) has only one root, its graph will have no turning points/extrema. If f(x) has a critical point, it must be a saddle point. Differentiating f(x) yields
[tex]f'(x) = 3x^2 - 2kx + 12[/tex]
Solve for the critical point:
[tex]f'(x) = 3x^2 - 2kx + 12 = 0[/tex]
[tex]x^2 - \dfrac{2k}3 x = -4[/tex]
[tex]x^2 - \dfrac{2k}3 x + \dfrac{k^2}9 = \dfrac{k^2}9-4[/tex]
[tex]\left(x - \dfrac k3\right)^2 = \dfrac{k^2}9-4[/tex]
[tex]x = \dfrac k3 \pm \sqrt{\dfrac{k^2}9-4}[/tex]
There is at most one real critical point, so either the square root term vanishes or it produces a non-real number. This happens for
[tex]\dfrac{k^2}9 - 4 \le 0 \implies k^2 \le 36 \implies -6 \le k \le 6[/tex]
So, if f(x) is one-to-one, then
[tex]k \in \left\{\kappa \in \Bbb R \mid -6 \le \kappa \le 6\right\}[/tex]
Related Questions
Look at the image below.
7
5
4
What is the area of the triangle?
Answers
Step-by-step explanation:
the area of a triangle is always
baseline × height / 2
where baseline and height are perpendicular (with a 90° angle) to each other.
it does not matter, if the height is inside or outside of the triangle.
so, in our case the area is
4×5/2 = 2×5 = 10 units²
The area of the triangle given in the figure is 10 units².
What is Area?Area of a two dimensional shape is the total region which is bounded by the object's shape.
Given a triangle.
The formula to find the area of the triangle is,
Area = 1/2 × base × height
Base = 4 units
Height = 5 units
Area of the triangle = 1/2 × 4 × 5
= 10 units²
Hence the area of the triangle is 10 units².
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A triangular pyramid is formed from three right triangles as shown below.
Use the information given in the figure to find the length FH.
If applicable, round your answer to the nearest whole number.
The lengths on the figure are not drawn accurately.
28
53
117
Answers
Answer:
FH = 108
Step-by-step explanation:
The given figure requires we use the Pythagorean theorem to write two relations involving right triangle side lengths. The Pythagorean theorem tells us the square of the hypotenuse is the sum of the squares of the other two sides.
__
Triangle EGH:EG² = GH² +HE²
GH² = EG² -HE² = 53² -28² = 2025 . . . . . solve for GH², use given values
Triangle FGH:FG² = GH² +FH²
FH² = FG² -GH² = 117² -2025 = 11664 . . . . solve for FH², use known values
FH = √11664 = 108 . . . . . take the square root
The length FH is 108.
A scale drawn on the map shows that 1 inch represents 20 miles. If two cities
are 2.5 inches apart on the map, what is the distance between them in real
life?
O A. 50 miles
O B. 20 miles
O C. 40 miles
D. 30 miles
Answers
Answer:
50 miles
Step-by-step explanation:
20 multiplied by 2.5 = 50
ayuda plis /help please
Answers
Answer:
a. 8
b. -12
c. 18
d. -22
e. 14
f. -2
g. -16
creo q seria asi no estoy muysegurappp
A shark spotter is standing at lookout point 25m above the waters edge. He spots a shark in the water at an angle of depression of 50. If the swimmer that is also in the water is sim from the foot of the lookout spot, how far is the shark from the swimmer?
Answers
The shark is 29.78 from the swimmer.
What is angle of depression?Angle of depression is the angle between the line of sight and the vertical or perpendicular to the earth surface.
Analysis:
The height of the lookout, the distance of shark from swimmer and line of sight form a right angle triangle.
So that, tan 50 = distance of shark from swimmer/25
Distance of swimmer = 25 tan 50
Distance of swimmer from shark = 25(1.1910 = 29.78m
In conclusion, the distance of the shark from the swimmer is 29.78m
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The distance of the shark from the swimmer is 21m
TrigonometryFrom the question, we are to determine the distance the shark is from the swimmer
In the diagram,
S is the position of the shark
M is where the shark spotter is standing
and B is the position of the swimmer
The distance between the shark and the swimmer is /BS/
Using SOH CAH TOA
[tex]tan 50^\circ = \frac{25}{/BS/}[/tex]
[tex]1.19175 = \frac{25}{/BS/}[/tex]
∴ [tex]/BS/ = \frac{25}{1.19175}[/tex]
/BS/ = 20.97755
/BS/ ≅ 21 m
Hence, the distance of the shark from the swimmer is 21m.
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Is 12 , 7 , 2 , -3 ... an arithmetic sequence ?
Answers
Answer:
Yes, you subtract 5 every time
Give a real-life situation for the sum. (will mark brainiest!!)
Answers
Answer:
Jeff is 15 years younger than Joe who is 10
Step-by-step explanation:
You might need:
Calculator
f(t) = 2 + 19t + 60
1) What are the zeros of the function?
Write the smaller t first, and the larger t second.
smaller t=
larger t =
2) What is the vertex of the parabola?
Answers
PLEASE HELP!!!! IMA DESPRATE!!!!!!!!Find the second term in the sequence defined by an = −5(n2 + 1n).
Answers
The second term of the given sequence aₙ = -5(n² + 1n) when solved gives us; a₂ = -20
How to find the nth term of a sequence?We are given the formula for a sequence as;
aₙ = -5(n² + 1n)
Where n is the position of the term.
Now, for the first term, we will have;
a₁ = -5(1² + 1(1))
a₁ = -10
The second term of the sequence is;
a₂ = -5(2² + 1(2))
a₂ = -20
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A frame contains 4 pounds of gold. Suppose gold is worth $1277 per ounce. How much would the gold in the frame be worth?
Answers
Answer:
81728
Step-by-step explanation:
1 libra tiene 16onz, entonces 16 onz* 1277 dolares =20432*4=81728
NO LINKS!!
VWXYZ ~ IJKGH. Find HI, GK, AND JK.
Answers
Answer:
HI = 35
GK = 45
JK = 25
Provided:
VWXYZ ≅ IJKGHGiven proportional sides:
GH = 30, YZ = 36Find the scale factor:
30/36 = 5/6Find missing sides:
1) HI = ZV × 5/6 = 42 × 5/6 = 35
2) GK = XY × 5/6 = 54 × 5/6 = 45
3) JK = WX × 5/6 = 30 × 5/6 = 25
Answer:
HI = 35
GK = 45
JK = 25
Step-by-step explanation:
If VWXYZ ~ IJKGH
Then:
VW ~ IJ WX ~ JKXY ~ KGYZ ~ GHZV ~ HIYZ and GH is the only pair of corresponding sides for which we have been given the lengths:
YZ = 36 and GH = 30
Therefore,
VWXYZ : IJKGH = 36 : 30 = 6 : 5
Find HI
ZV : HI = 6 : 5
⇒ 42 : HI = 6 : 5
⇒ 42/HI = 6/5
⇒ 42 · 5 = 6 HI
⇒ HI = 210/6
⇒ HI = 35
Find GK
YX : GK = 6 : 5
⇒ 54 : GK = 6 : 5
⇒ 54/GK = 6/5
⇒ 54 · 5 = 6 GK
⇒ GK = 270/6
⇒ GK = 45
Find JK
WX : JK = 6 : 5
⇒ 30 : JK = 6 : 5
⇒ 30 /JK = 6/5
⇒ 30 · 5 = 6 JK
⇒ JK = 150/6
⇒ JK = 25
A line passes through (- 1, 1) and (3,9).
What is the equation of the line in slope-intercept form?
O y = x + 6
O y = 2x + 3
O y = x + 2
O y = 2x + 9
Answers
Answer:
y = 2x + 3
Step-by-step explanation:
Hi there!
We are given the points (-1, 1) and (3,9) which belong to a line
We want to write the equation of the line in slope-intercept form
Slope-intercept form is given as y=mx+b, where m is the slope and b is the y intercept
First, let's find the slope of the line
The slope (m) can be found using the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are points
Let's label the values of the points before we start, to avoid confusion and mistakes
[tex]x_1 = -1\\y_1=1\\x_2=3\\y_2=9[/tex]
Now substitute into the formula (note: the formula has SUBTRACTION, and we have NEGATIVE numbers, so we'll end up subtracting a negative)
m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
m = [tex]\frac{9-1}{3--1}[/tex]
Simplify
m=[tex]\frac{9-1}{3+1}[/tex]
m=[tex]\frac{8}{4}[/tex]
Divide
m = 2
The slope of the line is 2
We can substitute this as m in y=mx+b.
y = 2x + b
Now we need to find b
As the equation passes through the point (-1, 1) and (3, 9), we can use either point to help solve for b.
Taking (3, 9) for example:
substitute 3 as x and 9 as y.
9 = 2(3) + b
Multiply
9 = 6 + b
Subtract 6 from both sides
3 = b
Substitute 3 as b in the equation
y = 2x + 3
Hope this helps!
Topic: finding the equation of the line
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The probability distribution for a
random variable x is given in the table.
X
-5 -3 -2 0
2 3
Probability .17
.13
.33 .16 .11
.10
Find the probability that x ≤ 0
Answers
Answer:
[tex]P(x\leq 0)=0.37[/tex]
Step-by-step explanation:
[tex]P(x\leq 0)= P(0)+P(2)+P(3)P(x\leq 0)= 0.16+0.11+0.10\\P(x\leq 0)=0.37[/tex]
Answer: 0.79
Step-by-step explanation:
Please help Khan Academy is due soon and I must FINISH!!!!!
Answers
When you distribute, you independently multiply each term inside the parentheses by the term outside the parentheses.
So
6 x 7 = 42
and
6 x -4n = -24n
So the final answer is: 42 - 24n
*You cannot combine this answer any further since you are subtracting an unknown variable (n) from a number.
I have this question and I can’t figure it out.
Answers
Answer:
-1. 7,-1,0,2,-2
Step-by-step explanation:
Intergers are from negative to positive numbers
help will mark Brainliest!!
Answers
Answer:
A. (h+(h-2) =14)...
I hope this is right! And go on your test! :)
Find the length of each arc. Round your answers to the nearest tenth.
5 m
285
Cuál es la respuesta???math
Answers
The length of the arc with a central angle of 285 degrees and a radius of 5 meters is 24.9 meters
How to determine the arc length?The given parameters are:
Central angle = 285Radius, r = 5 mThe length of the arc is calculated using:
[tex]L = \frac{\theta}{360} * 2\pi r[/tex]
So, we have:
L = 285/360 * 2 * 3.14 * 5
Evaluate
L = 24.9
Hence, the length of the arc is 24.9 meters
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Use the information in the table to find the indicated probability.
Number of Degree Recipients at a Local College
Degree Male Female
Associate's 224 387
Bachelor's 547 776
Advanced 245 322
Find P(the recipient is male).
Answers
Using it's concept, it is found that the desired probability is given as follows:
P(the recipient is male) = 0.4242 = 42.42%.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, there was a total of 224 + 387 + 547 + 776 + 245 + 322 = 2501 recipients, out of which 224 + 547 + 245 = 1016 are males, hence the probability is given by:
P(the recipient is male) = 1016/2501 = 0.4242 = 42.42%.
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• footballs – There are 8 footballs.
• basketballs – Two more than twice the number of footballs are
basketballs.
• baseballs – Four less than 5 times the number of footballs are
baseballs.
• softballs – Six more than half of the baseballs are softballs
Type of Ball
Numerical Expression for the Number of Balls
Total Number of
Each Type of Ball
football (given) 8
basketball
baseball
softball
Mainly need the last part (Numerical Expression for the Number of Balls)
Answers
Answer:
8 futballs
basketballs 18
beisballs 36
softballs 24
Step-by-step explanation:
dos mas del doble (2) de balones de fut = 8x2 = 16, 16+2 = 18
cuatro menos de 5 veces las pelotas de fut = 8x5 = 40, 40-4 = 36
seis mas de la mitad de beis = 36/2 = 18, 18+6 = 24
LISTO
which is a solution to x - 3x + 9 equals negative 27
Answers
Answer:
x = 18
Step-by-step explanation:
[tex]\sf x - 3x+9= - 27[/tex]
Let's first combine like terms.
[tex] \sf - 2x+9= - 27[/tex]
Subtract 9 from both sides.
[tex]\sf - 2x= - 27 - 9[/tex]
[tex] \sf - 2x= - 36[/tex]
Divide both sides by −2.
[tex]\sf x=18[/tex]
ABC School has 150 students enrolled who do not speak English. It is also known that 6/7 of the students can recognize all 26 letters. How many students still needs to learn the entire alphabet? Type your answer to the nearest whole number.
Answers
Answer:
about 21 students
Step-by-step explanation:
total students: 150 students enrolled
recognize all letters: 6/7 of the students
To find 1/7 of the students: divided the 150 by 7
150/7 = 21.4285 this is for 1/7th of students
since 1/7th of the students need to learn the alphabet (7/7 - 1/7 = 6/7)
OR
150/7 = 21.4285 this is for 1/7th of students
150/7 x 6 = 128.57 this is for 6/7th of students
150 - 128.57 = 21.4285
25x^4-8y^4 is not the difference of two squares. Identify the correct explanation for this statement.
Answers
Answer:
8y4 is not a perfect square.
Step-by-step explanation:
Knowing that; A difference of two perfect squares, A2 - B2 can also be factored into (A+B) • (A-B)
Verification;
(A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Notes;
AB = BA is the commutative property of multiplication.
AB + AB equals zero and is therefore eliminated from the expression.
Answer:
25 is the square of 5
8 is not a square
-15x+12
what is the answer
Answers
Answer:
If you need to factor this equation the answer is =3(-5x+4)
Step-by-step explanation:
-15x+12
-15/3 = -5
3/12 = 4
= 3(-5x+4)
Angle θ is in standard position. if sin(θ) = − 1/3, and π < θ < 3π/2 , find cos(θ).
A. -(2√2)/3
B.4/3
C.2√2/3
Answers
Answer:
Step-by-step explanation:
[tex]sin^2({ \theta})+cos^2({ \theta})=1\\cos({ \theta})=-/+\sqrt{1-sin^2({ \theta})} \\cos({ \theta})=-\sqrt{1-sin^2({ \theta})} \\ --- > sin({ \theta})=\frac{1}{3} (known)\\cos({ \theta})=-\sqrt{1-sin^2({ \theta})} =-\sqrt{1-(-\frac{1}{3})} =-\sqrt{\frac{9-1}{9}} =-\frac{2x^{2} }{3}[/tex]
please help me fast as you can
Answers
(i) 47 * 1 = 47
Why? This is because the Multiplication Identity Property states that any number w multiplied by 1 is equal to w
(ii) 135 * 0 = 0
Why? The Zero Property of Multiplication states that 0 multiplied by any number is 0.
(iii) 3 * 5 = 5 * 3
Why? The order numbers are multiplied in does not impact the answer's value according to the Commutative Property of Multiplication
(iv) 7 + 0 = 7
Why? The Zero Property of Addition states that 0 added to any number x is equal to x
(v) 125 * 315 - 125 * 215 = 125 * (315 - 215) = 12500
Why? The Distributive Property states that a * c - a * b = a * (c - b).
→
If a graph of y=-9x + 3 were changed to a graph of y=-9x + 1, how would the
y-intercept change?
A. The y-intercept of the new graph would shift up 2 units.
B. The y-intercept of the new graph would shift down 2 units.
C. More information is needed to determine the change.
D. The y-intercept would remain the same.
Answers
Answer:
B I think..................................
HELPPPPPPPPP MEEEE PLEASEEEEEE
Answers
Answer:
525 ml
Step-by-step explanation:
7 + 4 = 11 ml
825 ml ÷ 11 ml = 75 units for 1 part ( or A or B ) of drug
75 × 7 ml = 525 ml of part A
and
75 × 4 = 300 ml of part B
Use Pythagorean Theorem to solve for x. Show all work. Keep all answers in simplified
radical form.
Answers
Step-by-step explanation:
a^2 +b^2 = c^2
8^2 +4^2= c^2
64 + 16= c^2
80= c^2
take square root of both sides and you get
c=8.94
Find th areas and circumference 5 cm 2 cm
Answers
Answer:
The Area is 10cm, and the perimeter is 14cm
Step-by-step explanation:
Area:
a = w · l 5cm · 2cm =10cm
Perimeter:
p = 2(l+w) 2(5 + 2) = 14
I really need help on this one
Answers
Answer:
-3x+9y=-57
Substitute the x for 4
-3(4)+9y=-57
-12+9y=-57
Add +12
-12+9y=-57
+12 +12
9y=-45
Now divide the 9 to get y by itself
9y=-45
÷9 ÷9
y=-5
So the value of y is -5
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