The graph of f(x)= 3/1+x^2 is shown in the figure to the right. Use the second derivative of f to find the intervals on which f is concave upward or concave downward and to find the inflection points of f.
Answers
Answer:
Concave Up Interval: [tex](- \infty,\frac{-\sqrt{3} }{3} )U(\frac{\sqrt{3} }{3} , \infty)[/tex]
Concave Down Interval: [tex](\frac{-\sqrt{3} }{3}, \frac{\sqrt{3} }{3} )[/tex]
General Formulas and Concepts:
Calculus
Derivative of a Constant is 0.
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Quotient Rule: [tex]\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Chain Rule: [tex]\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Second Derivative Test:
Possible Points of Inflection (P.P.I) - Tells us the possible x-values where the graph f(x) may change concavity. Occurs when f"(x) = 0 or undefinedPoints of Inflection (P.I) - Actual x-values when the graph f(x) changes concavityNumber Line Test - Helps us determine whether a P.P.I is a P.IStep-by-step explanation:
Step 1: Define
[tex]f(x)=\frac{3}{1+x^2}[/tex]
Step 2: Find 2nd Derivative
1st Derivative [Quotient/Chain/Basic]: [tex]f'(x)=\frac{0(1+x^2)-2x \cdot 3}{(1+x^2)^2}[/tex]Simplify 1st Derivative: [tex]f'(x)=\frac{-6x}{(1+x^2)^2}[/tex]2nd Derivative [Quotient/Chain/Basic]: [tex]f"(x)=\frac{-6(1+x^2)^2-2(1+x^2) \cdot 2x \cdot -6x}{((1+x^2)^2)^2}[/tex]Simplify 2nd Derivative: [tex]f"(x)=\frac{6(3x^2-1)}{(1+x^2)^3}[/tex]Step 3: Find P.P.I
Set f"(x) equal to zero: [tex]0=\frac{6(3x^2-1)}{(1+x^2)^3}[/tex]Case 1: f" is 0
Solve Numerator: [tex]0=6(3x^2-1)[/tex]Divide 6: [tex]0=3x^2-1[/tex]Add 1: [tex]1=3x^2[/tex]Divide 3: [tex]\frac{1}{3} =x^2[/tex]Square root: [tex]\pm \sqrt{\frac{1}{3}} =x[/tex]Simplify: [tex]\pm \frac{\sqrt{3}}{3} =x[/tex]Rewrite: [tex]x= \pm \frac{\sqrt{3}}{3}[/tex]Case 2: f" is undefined
Solve Denominator: [tex]0=(1+x^2)^3[/tex]Cube root: [tex]0=1+x^2[/tex]Subtract 1: [tex]-1=x^2[/tex]We don't go into imaginary numbers when dealing with the 2nd Derivative Test, so our P.P.I is [tex]x= \pm \frac{\sqrt{3}}{3}[/tex] (x ≈ ±0.57735).
Step 4: Number Line Test
See Attachment.
We plug in the test points into the 2nd Derivative and see if the P.P.I is a P.I.
x = -1
Substitute: [tex]f"(x)=\frac{6(3(-1)^2-1)}{(1+(-1)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(1)-1)}{(1+1)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(3-1)}{(1+1)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(2)}{(2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(2)}{8}[/tex]Multiply: [tex]f"(x)=\frac{12}{8}[/tex]Simplify: [tex]f"(x)=\frac{3}{2}[/tex]This means that the graph f(x) is concave up before [tex]x=\frac{-\sqrt{3}}{3}[/tex].
x = 0
Substitute: [tex]f"(x)=\frac{6(3(0)^2-1)}{(1+(0)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(0)-1)}{(1+0)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(0-1)}{(1+0)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(-1)}{(1)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(-1)}{1}[/tex]Multiply: [tex]f"(x)=\frac{-6}{1}[/tex]Divide: [tex]f"(x)=-6[/tex]This means that the graph f(x) is concave down between and .
x = 1
Substitute: [tex]f"(x)=\frac{6(3(1)^2-1)}{(1+(1)^2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(3(1)-1)}{(1+1)^3}[/tex]Multiply: [tex]f"(x)=\frac{6(3-1)}{(1+1)^3}[/tex]Subtract/Add: [tex]f"(x)=\frac{6(2)}{(2)^3}[/tex]Exponents: [tex]f"(x)=\frac{6(2)}{8}[/tex]Multiply: [tex]f"(x)=\frac{12}{8}[/tex]Simplify: [tex]f"(x)=\frac{3}{2}[/tex]This means that the graph f(x) is concave up after [tex]x=\frac{\sqrt{3}}{3}[/tex].
Step 5: Identify
Since f"(x) changes concavity from positive to negative at [tex]x=\frac{-\sqrt{3}}{3}[/tex] and changes from negative to positive at [tex]x=\frac{\sqrt{3}}{3}[/tex], then we know that the P.P.I's [tex]x= \pm \frac{\sqrt{3}}{3}[/tex] are actually P.I's.
Let's find what actual point on f(x) when the concavity changes.
[tex]x=\frac{-\sqrt{3}}{3}[/tex]
Substitute in P.I into f(x): [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{1+(\frac{-\sqrt{3} }{3} )^2}[/tex]Evaluate Exponents: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{1+\frac{1}{3} }[/tex]Add: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{3}{\frac{4}{3} }[/tex]Divide: [tex]f(\frac{-\sqrt{3}}{3} )=\frac{9}{4}[/tex][tex]x=\frac{\sqrt{3}}{3}[/tex]
Substitute in P.I into f(x): [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{1+(\frac{\sqrt{3} }{3} )^2}[/tex]Evaluate Exponents: [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{1+\frac{1}{3} }[/tex]Add: [tex]f(\frac{\sqrt{3}}{3} )=\frac{3}{\frac{4}{3} }[/tex]Divide: [tex]f(\frac{\sqrt{3}}{3} )=\frac{9}{4}[/tex]Step 6: Define Intervals
We know that before f(x) reaches [tex]x=\frac{-\sqrt{3}}{3}[/tex], the graph is concave up. We used the 2nd Derivative Test to confirm this.
We know that after f(x) passes [tex]x=\frac{\sqrt{3}}{3}[/tex], the graph is concave up. We used the 2nd Derivative Test to confirm this.
Concave Up Interval: [tex](- \infty,\frac{-\sqrt{3} }{3} )U(\frac{\sqrt{3} }{3} , \infty)[/tex]
We know that after f(x) passes [tex]x=\frac{-\sqrt{3}}{3}[/tex] , the graph is concave up until [tex]x=\frac{\sqrt{3}}{3}[/tex]. We used the 2nd Derivative Test to confirm this.
Concave Down Interval: [tex](\frac{-\sqrt{3} }{3}, \frac{\sqrt{3} }{3} )[/tex]
Related Questions
Use an exponent to write each expression 3/4 x 3/4 x 3/4 x 3/4 x 3/4
Answers
Answer:
[tex] { \frac{3}{4} }^{5} [/tex]
The given expression 3/4 x 3/4 x 3/4 x 3/4 x 3/4 can be written as (3/4)⁵ in the form of an exponent.
What is the expression?Expressions are defined as mathematical statements that have a minimum of two terms containing variables or numbers.
An exponent is a shorthand way of representing a repeated multiplication of the same number.
As per the given question, the number being multiplied is 3/4, and it is being multiplied by itself 5 times. The exponent, 5, tells us how many times the base, 3/4, is being multiplied by itself.
⇒ 3/4 x 3/4 x 3/4 x 3/4 x 3/4
⇒ (3/4)⁵
Therefore, the given expression can be written as (3/4)⁵.
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Gabrielle is 10 years younger than Mikhail. The sum of their ages is 88. What is Mikhail's age?
Answers
Stu
The point on the parabola y = 3 - x2 that is closest to the point (3,3) is
(Type an ordered pair. Round to two decimal places as needed.)
Answers
Step-by-step explanation:
i!ieirurieiewjsisks
On a coordinate plane, the x-axis is labeled Number of Days and the y-axis is labeled Number of Computers. A line goes through points (3, 36), (4, 48), (5, 60). The graph shows a proportional relationship between the number of computers produced at a factory per day. In three days, 36 computers are produced; 48 computers are produced in 4 days; and 60 computers are produced in 5 days. Find the unit rate of computers per day using the graph. Unit rate: computers per day
Answers
Answer:
12 is the answer ppl hope it helps you all
Step-by-step explanation:
Answer:
12
Step-by-step explanation:
i got it right on edg
Anyone help with this
Answers
Answer:
Anyone help with this
x = 3t / bIf m∠ACD = 136° and m∠B = 42°, Find m∠A.
Answers
Answer:
2°
Step-by-step explanation:
Hello
I see you are very confused by this problem. However, there is an easy way to solve this one.
It has been proved that the sum of all angles in a rectangle is always 180°. Here is a way to prove it:
x y
_______A_________
/\
/ \
/ \
/ \
___/_______ \__
B C
we have two parellel lines xy and BC. It is true that:
xAy = 180°
xAB + BAC + CAy = xAy = 180
xAB = ABC
CAy= ACB
=> ........
I'll let you finish with your great intellect ^_^
HOPE YOU LEARN WITH JOY AND GOOD GRADE
1. Identify each part of the exponent: (-4)3
Base =
Exponent =
Answers
ThTs my answer I hope it’s correct
if you bought 5 apples for $2. What is the cost for 1 apple
Answers
Answer:
$0.40
Step-by-step explanation:
you divide the 2 by 5 because they two dollars is being spread throughout the 5 apples. 2 divided by 5 is 0.40
Steph bought a used bicycle restored it and then marked it up by 1/4 of the price she paid later she sold it for the marked-up price of $80 how much in dollar did steph pay for the bicycle ?
Answers
Answer:
Steph payed $64 for the bicycle
Step-by-step explanation:
80/5=16
16×4=64
hopefully this helps :)
Some of the line segments on the graph are nearly horizontal. What do these line segments represent?
Answers
Answer:
Step-by-step explanation:send a pic
Answer:
These line segments represent the end point of a graph
combine like terms 2x - 3 - 2y - 4x + 2y + 5
Answers
Answer:
-2x + 2
Step-by-step explanation:
2x - 3 - 2y - 4x + 2y + 5
2x - 4x + 2y - 2y + 5 - 3
-2x + 2
Jan tossed a coin 50 times and got a head 20 times. What is the
experimental probability? What is the theoretical probability?
Answers
Answer:
The standard answer for this is 50%.
Step-by-step explanation:
However, this is based on the implicit assumption that the coin is fair.
If there are reasonable grounds to doubt that the coin is fair, the theoretical probability must be based on observed statistics. In this scenario, the probability is 60%.
Tara is graphing the equation 4x + 2y = 10. Which of these shows the correct equation in slope-intercept form, slope, and y-intercept? Question 2 options: y = –2x + 5 slope = –2 y-intercept = 5 y = 4x + 5 slope = 4 y-intercept = 5 y = 2x + 4 slope = 2 y-intercept = 4 y = –4x + 5 slope = –4 y-intercept = 5
Answers
the answer should be : y=-2x+5
Some algae contains up to 50% oil by weight. How much oil is in a 7.3-ounce sample of this type of algae?
This would help if you could solve anyone.
Answers
Answer: 3.65
Step-by-step explanation:
This question is asking basically what is 50% or 1/2 of 7.3.
Let's expand the number 7.3
Then, we end up getting 7.30.
We have to find 1/2 of 7.30.
[tex]\frac{1}{2}[/tex] of 7.30 = [tex]3.65[/tex]
Hope this helps you!
425 divided by 14 please i need to know fast
Answers
Answer:
30.3571429
Step-by-step explanation:
Explanation: calculator
7(3x -4) = 49
can you help me
Answers
Answer:
3 2/3
Step-by-step explanation:
7(3x-4)=49
21x-28=49
21x=77
x=3 2/3
what are the common factors of 20 and 24
Answers
Answer:
4
Step-by-step explanation:
4 x 5= 20
4x6 = 24
So 4 can go in 20 and 24 making it the common factor
The common factor of 20 and 24 would be; 4
How to find Greatest common factor?Greatest common factor, as the nathe me implies, is greatest among all common factors among the quantities given. We can factorize the quantities in smallest relative factors possible (one level of factors over which all quantities can form themselves), and collect as many common factors as we could. Those will together greatest GCF(Greatest common factor).
We need to find the common factors of 20 and 24
Therefore,
Factors are;
4 x 5= 20
4 x 6 = 24
Hence, we can conclude that 4 can go in 20 and 24. it is the common factor.
The common factor of 20 and 24 would be; 4
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29-3x=5x+5
Can you work it out
Answers
Answer:
x=3
Step-by-step explanation:
29 - 3x = 5x + 5
Subtract 5 from both sides:
24 - 3x = 5x
Add 3x to both sides:
24 = 8x
Divide both sides by 8:
X = 3
4x - 4 - 8y + 8 - 5x + 1 =
Answers
Answer:
-x - 8y + 5
Step-by-step explanation:
Add the numbers
4x - 4 - 8y + 8 - 5x + 1 =
4x + 5 - 8y - 5y
Combine like terms
4x + 5 - 8y - 5y
-x + 5 - 8y
Rearrange terms
-x + 5 - 8y
-x - 8y + 5
The solution of expression is,
⇒ - x - 8y + 5
We have to given that,
An expression is,
⇒ 4x - 4 - 8y + 8 - 5x + 1
Now, We can simplify by combining like terms as,
⇒ 4x - 4 - 8y + 8 - 5x + 1
⇒ 4x - 5x - 8y - 4 + 8 + 1
⇒ - x - 8y + 5
Therefore, The solution of expression is,
⇒ - x - 8y + 5
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arange the number line in the following spending order 20,-40,5-1,10
Answers
I need help please Will mark brainliest. Thank you so much.
Answers
The factors of two numbers are given below: Number 1: 1, 2, 3, 6, 9, 18. Number 2: 1, 2, 3, 4, 6, 12. What is the least common multiple of the numbers?
Answers
Answer:
36 is your answer hope this helps
A park shaped like a rectangle. The diagram shows the how the park is split into a sports field and a playground. Let x represent the length in meters of the sports field. Write an expression to represent the area of the park as the sum of the areas of the sports field and the playground.
Answers
Answer:
20r + 310
Step-by-step explanation
For the first part.
The required expression is 20x + 310 which represents the area of the park as the sum of the areas of the sports field and the playground.
What is the expression?Expressions are defined as mathematical statements that have a minimum of two terms containing variables or numbers.
Let x represent the length in meters of the sports field.
Here, the area of the sports field = 20 × x = 20x units
And, the area of the playground = 15.5 × 20 = 310 units
Now, the required expression to represent the area of the park will be :
⇒ area of the sports field + area of the playground
⇒ 20x + 310
Thus, the required expression is 20x + 310.
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Please answer the screenshot I attached below
Answers
Answer:
i think the answer is
[tex] \frac{1}{ {x}^{ \frac{4}{3} }} [/tex]
Below are two parallel lines with a third line intersecting them.
Answers
Answer:
x = 132
Step-by-step explanation:
180 - 48 = 132
Help me with this please it says how can you isolate the term f/3 in f/3 + 22 = 17
Answers
Answer:
Given:[tex]\sf {\dfrac {f}{3}}+22=17 [/tex]
To find:-[tex]\sf {\dfrac {f}{3}}[/tex]
Solution:-First write the equation[tex]{:}\dashrightarrow[/tex][tex]\sf {\dfrac {f}{3}}+22=17 [/tex]
Now take 22 to the other side by which the "+"sign will turn "-".[tex]{:}\dashrightarrow[/tex][tex]\sf {\dfrac {f}{3}}=17-22 [/tex]
Simplify[tex]{:}\dashrightarrow[/tex][tex]\sf {\dfrac {f}{3}}=-5 [/tex]
[tex]\therefore[/tex][tex]{\underline{\boxed{\bf {\dfrac {f}{3}}=(-5)}}}[/tex]
Answer: Subtract 22 from both sides.
Step-by-step explanation:
f/3 +22 = 17
f/3 +22 -22 = 17 -22 Subtracting the same value from both sides keeps the equation equal.
Then simplify. The +22 and -22= 0 They "cancel" 17-22 = -5
f/3 = -17 f/3 is now isolated-- by itself-- in the equation.
If you had to solve for f, the next step would be to multiply both sides by 3.
9 ⋅ (-.5) what is the answer
Answers
Answer:
-45
Step-by-step explanation:
9x5=45, and since there is a negative, you put it before 45.
Hope I helped!
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A theater has 384 seats. Each row has 16 seats. The area model represents this situation.What is the value of x in the area model? Enter your answer in the box.
Answers
Answer:
X = 4 because 20 times 16 is 320 than if you add 4 to the 20 you will get 24. 24 times 16 is 384. BOom thats your answer and i know because i just answered it.
Step-by-step explanation:
At a local baseball game, the concession stand has two different meal choices. You can get 7 hotdogs and 5
drinks for $41.00 or you can get 4 hotdogs and 8 drinks for $44.00. If the price of hotdogs and drinks is the
same for each meal, find how much it costs for an individual hotdog and how much it costs for an individual
drink.
per hotdog:
per drink:
Answers
Answer:
Each hotdog is $3 and each drink is $4.
Step-by-step explanation:
Let h be the number of hotdogs and d the number of drinks,
7h + 5d = 41
4h + 8d = 44
Let's eliminate h. Multiply the first by 4 and the second by 7.
28h + 20d = 164
28h + 56d = 308
Subtrat the first from the second.
36d = 144
d = 4
Substitute back into one of the equations.
7h + 5(4) = 41
7h + 20 = 41
7h = 21
h = 3
Solve each of the following:
a. 3 movie tickets cost $36. At this rate, what is the cost per ticket?
b. 3 ice cream cones cost $8.25. At this rate, how
much do 2 ice cream cones cost?
C-3 bananas cost $0.99. At this rate, how much do 5 bananas cost?
a. $
b. $
Answers
Step-by-step explanation:
A. 36÷3=12 $12 per ticket
B. 8.25÷3=2.75 $2.75 per ice-cream cone so do 2.75×2=5.50 $5.50 for 2 ice creams
C. 0.99÷3=0.33 $0.33 per banana so do 0.33×5=1.65
$1.65 for 5 bananas
HOPE THIS HELPS