# f(x)=x+3x−−√f(x)=x+3x

**HISTORY**

Paul Revere was very famous for his role in the events leading to the Revolutionary War. As the colonists anticipated the movement of the British army out of Boston, they needed to be prepared to…

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**BUSINESS**

**Why are reserves considered as liability side on a balance sheet?**

You might think that the reserves that banks hold would be listed as assets on their books because this is money that they have in their possession. However, that is not the case. Instead,…

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**HISTORY**

**How did the Bronze Age cause a division in social classes?**

Though social class and stratification has its roots in the Agricultural Revolution, the Bronze Age significantly deepened the divides between social classes. Social stratification is fundamentally…

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**HISTORY**

**What is the purpose of the judicial branch deciding on laws?**

The purpose of the judicial branch deciding on the legality or constitutionality of laws is to ensure that a branch of government that did not create the laws is passing judgement on them. For…

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**HISTORY**

**Why did the colonists rebel against England?**

The colonists rebelled against England for several reasons. The colonists were upset with the tax laws that were passed. The colonists believed they should have a say about any tax law that would…

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**THE COLD EQUATIONS**

**What are examples of rising action in Tom Godwin’s short story “The Cold Equations”?**

In Tom Godwin’s short story “The Cold Equations,” pilot Barton’s internal conflict does not begin until the stowaway emerges to reveal herself to be a girl. Prior to that moment, there is no…

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**HISTORY**

**How did the US Constitution protect American liberties?**

If we are talking about the original US Constitution, not including the Bill of Rights or any other amendments, the answer is that it protected American liberties in two ways. First, the…

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**LAW AND POLITICS**

**The Bill of Rights was intended to limit the power of A. state governments over citizens B….**

The Bill of Rights, which is the first ten amendments of the United States Constitution, was intended to limit the power of the federal government. For example, the federal government was now…

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**JOHN LOCKE**

**How did theology impact John Locke?**

Although Locke is popularly known as a founder of empiricism whose works are strongly secular and rational in nature, he was not an atheist. Rather, Locke was a member in good standing of the…

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**REFERENCE**

**What is a way to describe someone who is cold hearted or with an ugly heart?**

I do not see that your question is attached to any particular work or book, so I will switch your topic to “reference”. The best way to find a synonym to describe something is by using a…

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**GOD’S GRANDEUR**

**What is the central idea of the poem?**

The central idea of “God’s Grandeur” is expressed in three parts–a sketch of God’s character, a lament and a psalm-like praise–and sets forth Hopkins’ view that God’s grandeur is apparent in the…

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**THE WHITE MAN’S BURDEN**

**What is the “exile” that Rudyard Kipling is talking about in the poem “White Man’s Burden?”**

Kipling says that Americans should “bind their sons to exile/To serve your captives’ need.” By “exile,” he means that America’s young men, especially its soldiers, ought to take up the challenge of…

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**AMERICAN HISTORY**

**What is an example of irony in Judith Ortiz Cofer’s short story “American History”?**

“American History” mostly centers around the day that President Kennedy got shot, which was November 22, 1963. The story has a number of ironic elements to it. First, Elena is a Puerto Rican…

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**SHOOTING AN ELEPHANT**

**What did George Orwell mean by the following sentence in his essay “Shooting an Elephant”: “It…**

George Orwell wrote “Shooting an Elephant” as, what some have said, is an autobiographical essay, or at least semi-autobiographical. George Orwell spent time in Burma (Myanmar) as a member of the…

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**THE CORSICAN BROTHERS**

**What were the family expectations in The Corsican Brothers by Alexandre Dumas père?**

The Corsican Brothers is a novella by Alexandre Dumas, père, (1802-1870) detailing a visit by the narrator to Sicily and meeting with one of two twin brothers and his return to Paris where he…

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**SCIENCE**

**Can you form a molecule by bonding 2 oxygen atoms and 1 hydrogen atom together?**

Yes. A molecule does exist that is composed of two oxygen atoms and one hydrogen atom bonded together. The substance is called hydroperoxyl radical and can be represented as HO2. Covalent bonds…

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**LAMB TO THE SLAUGHTER**

**Even though the wife murders her husband, the readers find themselves rooting for her to get away…**

There are several reasons why the reader ends up rooting for the wife in the story after she murders her husband. The most obvious way to argue in favor of the wife is to look at the story from a…

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**THE METAMORPHOSIS**

**What significance is attached to the food in The Metamorphosis?**

Throughout the story, “Metamorphosis,” the food is used as a symbol to represent several concepts. However, some of the most apparent and vital uses are the food’s symbol of transformation…

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**LETTER FROM BIRMINGHAM CITY JAIL**

**What ideas and emotions in the letter had the greatest impact on you?**

Martin Luther King’s August 1963 Letter from a Birmingham Jail certainly evinces the discipline and intellect which this leader possessed; indeed, he stands in great contrast to some who would call…

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**SCIENCE**

**What is the relationship between kinetic energy, heat and temperature?**

Heat, temperature and kinetic energy are linked to each other. In simplest terms, when we heat a substance, its temperature rises and causes an increase in the kinetic energy of its constituent…

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**LITERATURE**

**What is the summary for “The Guest Worker Question in Postwar Germany” by Rita Chin?**

In The Guest Worker Question in Postwar Germany, Rita Chin speaks very specifically about Germany after World War II. This eNotes Educator is reminded heavily of the visit I paid to Berlin, Germany…

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**SCIENCE**

**An electron moving along the x-axis has a position given by x = Ate – Bt m, where t is in seconds…**

Position of the electron is x = Ate – Bt m (along the x-axis). A and B are positive constants. When the electron stops, its velocity is 0. Since we are given the position, we can convert it to…

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**ESSAY LAB**

**I need to introduce three social topics that I am interested in and explain why they are of…**

The following are some tips to help you get started on your paper topic:Sociology is the study of human interactions through relationships and of the institutions humans have created. Sociologists…

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**BLUES AIN’T NO MOCKINGBIRD**

**What is the message (topic/theme) that the writer is trying to convey?**

The central theme of “Blues Ain’t No Mockingbird” by Toni Cade Bambara is respect for others, including their social and emotional situations. A two-man crew takes pictures around the…

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**A RETRIEVED REFORMATION**

**In “A Retrieved Reformation” what were Jimmy’s successes in Elmore?**

When Jimmy Valentine descends from the “mail-hack” in Elmore, Arkansas, he soon becomes “another man”: He successfully opens a shoe store and gains the love of Annabel Adams, the approval of her…

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**MATH**

Given the function f(x)=2sinx+cos(2x) in the interval [0,2π][0,2π] Taking the first derivative we get, f'(x)=2cosx-2sin(2x) Inorder to find the critical points we have to equate f'(x)=0. So we get,…

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**MATH**

**f(x)=cos(x)−x,[0,4π]f(x)=cos(x)-x,[0,4π] Find all relative extrema, use the second derivative test…**

Given the function f(x)=cosx−xf(x)=cosx-x in the interval [0,4π][0,4π] Now taking the derivative we get, f'(x)=−sinx−1f′(x)=-sinx-1 To find the critical points, we have to equate f'(x)=0 therefore, -sinx-1=0…

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**MATH**

**f(x)=xx−1f(x)=xx-1 Find all relative extrema, use the second derivative test where applicable.**

First derivative can be found using the quotient rule. f'(x)=(x−1)⋅1−x⋅x(x−1)2f′(x)=(x-1)⋅1-x⋅x(x-1)2 f'(x)=−x2+x−1(x−1)2f′(x)=-x2+x-1(x-1)2 At critical points, f'(x) = 0 -x^2+x-1=0 Roots can be found by the using the…

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**MATH**

**f(x)=x+4xf(x)=x+4x Find all relative extrema, use the second derivative test where applicable.**

First derivative, f'(x)=1+4⋅(−1)⋅x−1−1=1−4x−2f′(x)=1+4⋅(-1)⋅x-1-1=1-4x-2 At critical points, f'(x)=0,f′(x)=0, 1−4×2=01-4×2=0 1=4×21=4×2 x2=4×2=4 x=−2orx=+2x=-2orx=+2 At x = -2, f(x) = -2+4/(-2) = -4. At x = 2, f(x) =…

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**MATH**

**f(x)=x2+1−−−−−√f(x)=x2+1 Find all relative extrema, use the second derivative test where…**

First derivative, f'(x)=12⋅2⋅x(x2+1)12−1f′(x)=12⋅2⋅x(x2+1)12-1 f'(x)=x(x2+1)−12f′(x)=x(x2+1)-12 Therefore, x(x2+1)−12=0.x(x2+1)-12=0. This gives, x = 0 At x = 0, f(x) = 1. Therefore, the point (0,1) is a critical…

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**MATH**

**f(x)=x23−3f(x)=x23-3 Find all relative extrema, use the second derivative test where…**

First derivative, f'(x)=23⋅x23−1−0=23x−13f′(x)=23⋅x23-1-0=23x-13 At critical points, f'(x)=0f′(x)=0 Therefore, 23x−13=023x-13=0 This gives, x = 0 At x=0,f(x)=0−3=−3.x=0,f(x)=0-3=-3. Therefore, the point (0,-3)…

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**MATH**

**f(x)=−x4+4×3+8x2f(x)=-x4+4×3+8×2 Find all relative extrema, use the second derivative test where…**

Given: f(x)=−x4+4×3+8x2f(x)=-x4+4×3+8×2 Find the critical x values by setting the first derivative equal to zero and solving for the x value(s). f'(x)=−4×3+12×2+16x=0f′(x)=-4×3+12×2+16x=0 −4x(x2−3x−4)=0-4x(x2-3x-4)=0 −4x(x−4)(x+1)=0-4x(x-4)(x+1)=0…

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**MATH**

**f(x)=x4−4×3+2f(x)=x4-4×3+2 Find all relative extrema, use the second derivative test where…**

The first derivative, f'(x)=4⋅x3−4⋅3⋅x2+0=4×3−12×2=4(x3−3×2)f′(x)=4⋅x3-4⋅3⋅x2+0=4×3-12×2=4(x3-3×2) At critical points, f'(x)=0f′(x)=0 4(x3−3×2)=04(x3-3×2)=0 x2(x−3)=0x2(x-3)=0 This gives, x=0orx=3x=0orx=3 At x=0,f(x)=2x=0,f(x)=2 and, at…

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**MATH**

**f(x)=−x3+7×2−15xf(x)=-x3+7×2-15x Find all relative extrema, use the second derivative test where…**

- I) Find the first derivative -> f'(x) = -3x^2 + 14x – 15 and set it equal to zero to solve. This will find the x-values of your “critical points”. A critical point is where the graph humps….

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**MATH**

**f(x)=x3−3×2+3f(x)=x3-3×2+3 Find all relative extrema, use the second derivative test where…**

First derivative to locate the x coordinates of critical points. f'(x)=3⋅x2−3⋅2⋅x+0=3×2−6xf′(x)=3⋅x2-3⋅2⋅x+0=3×2-6x At critical points, f'(x)=0,f′(x)=0, 3×2−6x=03×2-6x=0 x2−2x=0x2-2x=0 x(x−2)=0x(x-2)=0 This gives,

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**MATH**

**f(x)=x2+3x−8f(x)=x2+3x-8 Find all relative extrema, use the second derivative test where…**

f=x2+3x−8f=x2+3x-8 First derivative to find the locations of extreme points, f’=2⋅x+3−0=2x+3f′=2⋅x+3-0=2x+3 At extremes points, f’=0,2x+3=0f′=0,2x+3=0 therefore, x=−32x=-32 Now, we know that there is an extreme…

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**MATH**

**f(x)=6x−x2f(x)=6x-x2 Find all relative extrema, use the second derivative test where applicable.**

f=6x−x2f=6x-x2 f’=6−2⋅xf′=6-2⋅x f’=2(3−x)f′=2(3-x) At extrema, f’=0, this give 3−x=0∴x=33-x=0∴x=3 At x = 3, there is a local extreme point. At x = 3, f=6⋅3−32=18−9=9f=6⋅3-32=18-9=9 Second derivative…

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**MATH**

**f(x)=x+2cos(x)[0,2π]f(x)=x+2cos(x)[0,2π] Find the points of inflection and discuss the concavity of…**

Find the inflection points and discuss the concavity for the function f(x)=x+2cos(x) on the interval [o,2pi]: Inflection points are found when the second derivative is zero (and has changed sign.)…

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**MATH**

Given f(x)=2sin(x)+sin(2x) on the interval [0,2pi], find the points of inflection and discuss the concavity: Find the second derivative: f'(x)=2cos(x)+2cos(2x) f”(x)=-2sin(x)-4sin(2x)…

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**MATH**

This function is fully differentiable on a given interval. We can use the second derivative to determine concavity and inflection points. Let’s find f” and where its positive, negative or zero:…

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**MATH**

**f(x)=sec(x−π2),(0,4π)f(x)=sec(x-π2),(0,4π) Find the points of inflection.**

You need to evaluate the points of inflection of the function, hence, you need to solve for x the equation f”(x) = 0. First, you need to evaluate the first derivative, using chain rule, such…

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**MATH**

**f(x)=2csc(3×2),(0,2π)f(x)=2csc(3×2),(0,2π) Find the points of inflection.**

f(x)=2csc(3×2)f(x)=2csc(3×2) differentiating, f'(x)=−2csc(3×2)cot(3×2)⋅(32)f′(x)=-2csc(3×2)cot(3×2)⋅(32) f'(x)=−3csc(3×2)cot(3×2)f′(x)=-3csc(3×2)cot(3×2) differentiating again, f”(x)=−3(csc(3×2)ddxcot(3×2)+cot(3×2)ddxcsc(3×2))f′′(x)=-3(csc(3×2)ddxcot(3×2)+cot(3×2)ddxcsc(3×2))…

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**MATH**

This function is defined everywhere and is infinitely differentiable. At the points of inflection of f the second derivative must be zero: f'(x)=(1/2)*cos(x/2), f”(x)=-(1/4)*sin(x/2). It is =0 at…

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**MATH**

**f(x)=x+3x−−√f(x)=x+3x Find the points of inflection and discuss the concavity of the…**

Given the function f(x)=x+3x−−√f(x)=x+3x i.e it can be written as f(x)=xx−−√+3x−−√f(x)=xx+3x =x12+3x−12×12+3x-12 Taking the first derivative we get, f'(x)=12x−12−32x−32f′(x)=12x-12-32x-32 Now again…

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**MATH**

**f(x)=4×2+1f(x)=4×2+1 Find the points of inflection and discuss the concavity of the graph of…**

Given the function f(x)=4×2+1f(x)=4×2+1 Taking the first derivative we get, f'(x)=−8x(x2+1)2f′(x)=-8x(x2+1)2 Again differentiating we get, f”(x)=−8(x2+1)2+8x(4×3+4x)(x2+1)4f′′(x)=-8(x2+1)2+8x(4×3+4x)(x2+1)4…

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**MATH**

**f(x)=x9−x−−−−−√f(x)=x9-x Find the points of inflection and discuss the concavity of the graph…**

Find the inflection points and discuss the concavity of the function f(x)=x⋅9−x−−−−−√f(x)=x⋅9-x The inflection points occur when the second derivative is zero (and changes sign.) A function’s graph is…

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**MATH**

**f(x)=xx+3−−−−−√f(x)=xx+3 Find the points of inflection and discuss the concavity of the graph…**

Find the inflection points and discuss the concavity for the function f(x)=xx+3−−−−−√f(x)=xx+3 Inflection points occur where the second derivative is zero (and changes sign.) Rewrite as

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**MATH**

**f(x)=(x−2)3(x−1)f(x)=(x-2)3(x-1) Find the points of inflection and discuss the concavity of the…**

f is infinite differentiable and its concavity is fully described by the sign of f”. f'(x)=3(x−2)2⋅(x−1)+(x−2)3=f′(x)=3(x-2)2⋅(x-1)+(x-2)3= (x−2)2⋅(3x−3+x−2)=(x−2)2⋅(4x−5).(x-2)2⋅(3x-3+x-2)=(x-2)2⋅(4x-5).

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**MATH**

**f(x)=x(x−4)3f(x)=x(x-4)3 Find the points of inflection and discuss the concavity of the graph of…**

f(x) is concave up First derivative, f'(x)=1⋅(x−4)3+x⋅3⋅1⋅(x−4)2f′(x)=1⋅(x-4)3+x⋅3⋅1⋅(x-4)2 f'(x)=(x−4)2(x−4+3x)f′(x)=(x-4)2(x-4+3x) f'(x)=4(x−4)2(x−1)f′(x)=4(x-4)2(x-1) There are only two critical points, x = 4 and x = 1 Second derivative,…

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**MATH**

You need to find the inflection points, hence, you need to solve for x the equation f”(x)=0f′′(x)=0 . First, you need to determine the first derivative:

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