The points of inflection for the function [f(x) = (x⁴ - 6x²)] are (1, -5) and
(-1, -5).
As per the question statement, we are provided with a function
[f(x) = (x⁴ - 6x²)].
We are required to determine the points of inflection for the above mentioned function.
To solve this question, we will have to calculate the second derivative of the given function. Then, we will have to equate the derivative to zero and solve for x. The obtained values for "x" will be our abscissa, and substituting these values of "x" in f(x), we will obtain our respective ordinates [Since, {y = f(x)}].
[tex]f(x)=(x^{4} -6x^{2})\\or,\frac{d[f(x)}{dx} = \frac{d(x^{4} -6x^{2}) }{dx}\\ or, f'(x)=\frac{dx^{4}}{dx}-\frac{d(6x^{2})}{dx}\\ or, f'(x)=\frac{dx^{4}}{dx}-6\frac{d(x^{2})}{dx}\\or, f'(x)=4x^{(4-1)}-(6*2)[x^{(2-1)}]\\ or,f'(x)=4x^{3}-12x\\[/tex]
Therefore, the second derivative of f(x) = [tex]\frac{d^{2}f(x)}{dx^{2} }[/tex]
[tex]or,f"(x)=\frac{d[f'(x)]}{dx}\\or,f"(x)=\frac{d(4x^{3} -12x)}{dx}\\or, f"(x)=\frac{d(4x^{3}) }{dx}-\frac{d(12x)}{dx}\\ or, f"(x)=4\frac{d(x^{3}) }{dx}-12\frac{d(x)}{dx}\\or,f"(x)=(4*3)[x^{(3-1)}]-(12*1)[x^{(1-1)}]\\ or, f"(x)=12x^{2} -12[/tex]
Now, equating (12x² - 12) to Zero, we get,
[tex](12x^{2} -12)=0\\or,12x^{2} =12\\or,x^{2} =1\\or,x=(1, -1)[/tex]
Now, substituting (x = 1) in f(x), we get,
[tex]y=f(1)=[(1)^{4}-6(1)^{2}]\\ or,y=[1-(6*1)]\\or,y=(1-6)\\or,y=(-5)[/tex]
Hence, one point of inflection for the function [f(x) = (x⁴ - 6x²)] is (1, -5).
Again, substituting [x = (-1)] in f(x), we get,
[tex]y=f(-1)=[(-1)^{4}-6(-1)^{2}]\\ or,y=[1-(6*1)]\\or,y=(1-6)\\or,y=(-5)[/tex]
That is, another point of inflection for the function
[f(x) = (x⁴ - 6x²)] is (-1, -5).
Inflection Point (Of a Function): In differential calculus and differential geometry, an inflection point, point of inflection or flex, is that point on a smooth plane curve at which the curvature changes its concavity i.e. from being "concave up" to being "concave down" or vice versa, or if stated in simpler terms, that point where the curve changes its sign.To learn more about Points of Inflection, click on the link below.
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A package of 15 batteries cost $14.25 what is the cost per battery
Answer: $0.95 per battery
Step-by-step explanation: Divide 14.25 and 15.
Simplifyyyyyyyyyyyyyyyyy
Answer:
Here ya go:
Step-by-step explanation:
Answer:
ok, yea, the above answer is correct
Step-by-step explanation:
Solve the quadratic equation by completing the square.
Answer:
x'1 = -1
x'2 = 8
Step-by-step explanation:
Answer:
x = - 1, x = 8
Step-by-step explanation:
Given
x² - 7x - 8 = 0 ( add 8 to both sides )
x² - 7x = 8
To complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2(- [tex]\frac{7}{2}[/tex] )x + [tex]\frac{49}{4}[/tex] = 8 + [tex]\frac{49}{4}[/tex] = [tex]\frac{81}{4}[/tex]
(x - [tex]\frac{7}{2}[/tex] )² = [tex]\frac{81}{4}[/tex] ( take the square root of both sides )
x - [tex]\frac{7}{2}[/tex] = ± [tex]\sqrt{\frac{81}{4} }[/tex] = ± [tex]\frac{9}{2}[/tex] ( add [tex]\frac{7}{2}[/tex] to both sides )
x = [tex]\frac{7}{2}[/tex] ± [tex]\frac{9}{2}[/tex]
Thus
x = [tex]\frac{7}{2}[/tex] - [tex]\frac{9}{2}[/tex] = [tex]\frac{-2}{2}[/tex] = - 1
x = [tex]\frac{7}{2}[/tex] + [tex]\frac{9}{2}[/tex] = [tex]\frac{16}{2}[/tex] = 8
The volume of a glass container is given to be 2x3+9x2-11x-30. If the height of the glass
container is given to be x - 2, what are the dimensions of the rectangular base of the glass
container?
Answer:
d
Step-by-step explanation:
if ummm its d
Which graph represents the function fx) =|x–4?
The image below shows the graph f(x) = |x| – 4
Hope this helps! :)
100 POINTS!
Factor [tex]\frac{7x^{2} + 11x -6}{x^{2}-4}[/tex]
Show your work, please!
Answer:
Following is the required answer for your question.
Thank you!
BRAINLIEST IF YOU ANSWER
Answer:
34.5 ft squared
Step-by-step explanation:
Split the figure into a triangle and two rectangles. The two vertical lines on the figure already do that. The triangle's area is 3*3/2 because 8 - 3 - 2 = 3 is the base and the height is 3. Use b * h/2. The area of the smaller rectangle is 2 *3. The area of the larger rectangle is 8 * 3. Add the areas up: 4.5 + 6 + 24 = 34.5.
pls help worth a lot of points
Answer: Ordinarily, no, it wouldn't matter.
Step-by-step explanation: No matter in what order you add it up it will be the same answer in the end.
Mayfield Water Company manufactures two types of bottled water: regular and flavored. The company produces at least twice as many bottles of regular water as bottles of flavored water. The company can produce, at most, 1,500 bottles of water per week. Which of these combinations would be reasonable for the company to produce in one week’s time?
A. 750 flavored and 750 regular
B. 500 flavored and 1,000 regular
C. 750 flavored and 1,000 regular
D. 1,000 flavored and 2,000 regular
Answer:
the answer is A
Step-by-step explanation: it's a because if u add 750 + 750 it will give you 1,500
the question: a computer manufacturer built a new facility for assembling computers. there were construction and new equipment cost. the company paid for these cost and made combined profits of $80 million after four years as shown in the graft.
answers:
a: f(x)=60x-160
b:f(x)=-60x+160
c:f(x)=-80x +4
d: f(x)= 80x -4
Answer:
The equation that closely represents the graph is [tex]\mathbf{y=60x-160}[/tex]
Option A is correct.
Step-by-step explanation:
We need to find equation that represents the graph.
We can write equation in slope-intercept form: [tex]y=mx+b[/tex] where m is slope and b is y-intercept.
For finding the equation we need to find slope and y-intercept
Finding slope
The slope can be found using formula: [tex]Slope=\frac{y_2-y_1}{x_2-x_1}[/tex]
From the graph we can find [tex]x_1=0, y_1=-160, x_2=1, y_2=-100[/tex]
Putting values and finding slope
[tex]Slope=\frac{y_2-y_1}{x_2-x_1}\\Slope=\frac{-100-(-160)}{1-0} \\Slope=\frac{-100+160}{1}\\Slope=-100+160\\Slope=60[/tex]
We get Slope = 60
Finding y-intercept
Using point (0,-160) and slope m = 60, we can find y-intercept
[tex]y=mx+b\\-160=60(0)+b\\b+0=-160\\b=-160[/tex]
Writing the equation
So, the equation for given graph having slope m = 60 and y-intercept b =-160 is:
[tex]y=mx+b\\y=60(x)+(-160)\\y=60x-160[/tex]
So, The equation that closely represents the graph is [tex]\mathbf{y=60x-160}[/tex]
Option A is correct.
Graph the image of this triangle after a dilation with a scale factor of 1/2 centered at (−5, 1).
Use the Polygon tool to graph the dilated triangle.
Answer:
Please see picture below.
Step-by-step explanation:
Pls graph and answer!
Answer:
Closed bracket “ ] “
at -5 facing the left then fill in the graph from -5 to -10
It’s facing left because -5 is greater than -x-4
Answer:
x ≥ 1
Step-by-step explanation:
Move all terms not containing x to the right side of the inequality.
−x ≤ −1
Divide each term in −x ≤ −1 by −1 and simplify.
x ≥ 1
Hope this helps! :)
4y-7-(y-7) find answer
Answer:
3y
Step-by-step explanation:
4y-7-1(y-7)
4y-7-y+7
4y-y
3y
. Vacation Days If the average number of vacation days for a selection of various countries has a mean of 29.4 days and a standard deviation of 8.6, find the z scores for the average number of vacation days in each of these countries.
Answer:
For Canada = -0.3953
For Italy = 1.46511
For USA = -1.9069
Step-by-step explanation:
Given:
Mean μ = 29.4 days
Standard deviation σ = 8.6
Missing Data
For
Canada = 26 days
Italy = 42 days
USA = 13 days
Find:
Z scores
Computation:
Z scores = [X-μ]/σ
For Canada
Z scores = [26-29.4]/8.6
Z scores = -0.3953
For Italy
Z scores = [42-29.4]/8.6
Z scores = 1.46511
For USA
Z scores = [13-29.4]/8.6
Z scores = -1.9069
The force required to compress a spring varies directly as the change in the length of the spring. If a force of 15 pounds is required to compress a certain spring 5 inches, how much force is required to compress the spring 8 inches?
The equation will be F = 3x. Then the force required to compress the spring 8 inches will be 24 pounds.
What are ratio and proportion?A ratio is a collection of ordered integers a and b represented as a/b, with b never equaling zero. A proportionate expression is one in which two items are equal.
A spring's length directly relates to the amount of force needed to compress it.
Let F be the force and x be the compressed length. Then the equation will be
F ∝ x
F = kx
If a force of 15 pounds is required to compress a certain spring 5 inches. Then the value of the variable k will be
15 = 5k
k = 15/5
k = 3 pounds per inch
Then the equation will be
F = 3x
The force required to compress the spring 8 inches will be
F = 3 (8)
F = 24 pounds
The force required to compress the spring 8 inches will be 24 pounds.
More about the ratio and the proportion link is given below.
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Use the original price and the markup to find the retail price.
Original price: $60; Markup: 20%
Answer:
$72 would be the retail price
Step-by-step explanation:
A football field is 300ft long a sloth moving very quickly travels 60ft every 5 minutes based on this ratio how many minutes would it take a sloth to travel the length of a football field
Answer:
25 minute
Step-by-step explanation:
Speed is the ratio of the total distance traveled to the total time taken to reach this distance. The formula for speed is:
Speed = total distance / total time taken
The total distance covered by the sloth = 60 ft, and the time taken = 5 minutes, hence:
Speed = total distance / total time taken
speed of the sloth = 60 ft / 5 minutes = 12 ft per minute.
We can calculate the time taken for the sloth to travel the length of the football field (i.e. 300 ft)
speed = distance / time
12 ft/minute = 300 ft / time
time = 300 ft / (12 ft/minute)
time = 25 minute
In electronics, the formula V = IR is called Ohm's Law. It gives the relationship in a circuit between the voltage V (in volts), the current I (in amperes), and the resistance R (in ohms). Find R when I = 2 − 3i amperes and V = 34 + i volts.
Answer:
R = 5 + 8i (Ohm)
Step-by-step explanation:
R = V/I
= (34 + i)/(2 - 3i)
= 5 + 8i (Ohm)
Can you explain this question please??
Answer:
the answer is here.
Step-by-step explanation:
hope it will helps you
Simplify the expression. Explain each step 7(4y)
Answer:
28y
Step-by-step explanation:
7 x 4 =28
then just add the y to it
and you get
28y
Theoretically, if the spinner is spun 150 times, how many times would you expect to get an even number?
150/2 = 75
The spinner would have to be spun 75 times to expect to get an even number.
Which shows the correct solution of the equation 1/2a+2/3b=50 , when b=30
The solution to the equation 1/2a + 2/3b = 50 when b = 30 is a = 60
How to determine the correct solution of the equation?The equation is given as
1/2a + 2/3b = 50
The value of b is given as
b = 30
So, we have
1/2a + 2/3 * 30 = 50
Evaluate the product
So, we have
1/2a + 20 = 50
Evaluate the like terms
So, we have
1/2a = 30
Multiply both sides by 2
a = 60
Hence, the solution to the equation 1/2a + 2/3b = 50 when b = 30 is a = 60
Read more about equations at
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Write an expression, using an exponent, that is equivalent to 8×8×8×8
Answer:
8 to the power of 4
Step-by-step explanation:
since their are four 8, (8x8x8x8) , you have to put that into an exponent
still need proof?
8x8x8x8=4096
8 to the power of 4 = 4096
so to conclude, 8 to the power of 4 is the same thing as saying 8x8x8x8, it is jsut the exponent version
HELP PLS. Probability. In a game a spinner will land on red 35% of the time, yellow 40% of the time, and blue 25% of the time. If the spinner is spun three times, find the following probabilities.
Can I get help on 1 and 2
Answer:
1.the slope is 1/2 and the y int is -4
2.the slope is -3 and the y int is 6
Explain how to find 40x80 using mental math
so 40 * 80 = 3,200
4 * 8 = 32
Then add the two 0's to the end
so 3,200
Answer:
The best way to solve 40x80 using mental math is to break down the problem.One example would be to solve 4x8 then add the two zeros to your answer to get 3200.
Step-by-step explanation:
2 tablets, 3x per day for 10 days
Answer:
60
Step-by-step explanation:
Simple
First you multiply 2 x 3 =6
Then you multiply 6 x 10 =60
That mean 60 is the answer you are looking for
Hope This Helps :)
Given two objects represented by the tuples (22, 1, 42, 10) and (20, 0, 36, 8):(a)Compute the Euclidean distance between the two objects. (b) Compute the Manhattan distance between the two objects.
Answer:
(a) 6.708
(b) 11
Step-by-step explanation:
Tuples ( 22, 1, 42, 10) and (20, 0, 36, 8)
Euclidian distance formula
[tex]D = \sqrt{(q1 -p1)^2 + (q2-p1)^2 + (q3 - p3)^2 + (q4 -p4)^2}[/tex]
D = [tex]\sqrt{(22-20)^2 + (1-0)^2 + (42 -36)^2 + (10 -8)^2 }[/tex]
= [tex]\sqrt{(2)^2 + (1)^2 + (6)^2 + (2)^2}[/tex]
= [tex]\sqrt{4 + 1 + 36 + 4 }[/tex]
= [tex]\sqrt{45}[/tex]
D = 6.708
Therefore the Euclidean distance between the tuples is 6.708
Manhattan Distance formula
D = |q1-p1| + |q2 -p2| + |q3 -p3| + |q4-p4|
= |22-20| + | 1-0| + | 42 -36| + | 10-8|
= |2| + |1| + |6| + |2|
D = 11
Therefore the Manhattan distance is equal to 11
A shirt was marked down 25% which means you saved $15. What was the original cost of the shirt? What is the new price?
Justin's new music player has more than 50 songs
but less than 90 songs. The number of songs is 3
more than a multiple of 5. It is also 2 more than a
multiple of 6. How many songs are on the player?