# f(x)=x−2ln(x)f(x)=x-2ln(x) Find the critical numbers of the function

**LITERATURE**

**What is a good way to study and prepare for a test on characterization in “The Scarlet Ibis?”**

There are several ways an author lets us get to know and understand a character in a literary work. We get to know characters by: The things the characters say The character’s actions (the things…

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**THE MONKEY’S PAW**

The author does not spell out Mr. White’s third and last wish, but the reader can easily imagine what the old man wished for. He wished for the knocking to stop and for the knocker to go away…

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**TO KILL A MOCKINGBIRD**

**Briefly list the things that Scout and Jem find in the knothole of the tree. Which items do Scout…**

Here is a list of the items that Jem and Scout find in the knothole of a tree. A gray ball of twine Two small images carved out of soap, which also resembles Scout and Jem A pack of gum An old…

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**THE SCARLET IBIS**

**If Brother were a season, he would be _______________ because _________________________.**

Because the main conflict of the story Brother is telling about his brother, Doodle, takes place in the autumn season, perhaps it best represents Brother’s time period in his life when he is…

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**SINNERS IN THE HANDS OF AN ANGRY GOD**

**What is a good quote in the sermon “Sinners in the Hands of an Angry God” for ethos?**

It is not surprising that you would have trouble finding examples of ethos in “Sinners in the Hands of an Angry God.” This sermon is based much more on pathos than on ethos. Edwards does not…

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**RAYMOND’S RUN**

**How can a reader say for sure that Squeaky is serious about running?**

To Squeaky, running is a passion. Besides looking after her elder brother Raymond, it is her most important activity. She’s got no particular time to go running. Instead she’s always running or…

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**BOYS AND GIRLS**

**What is a thematic topic in “Boys and Girls” by Alice Munro?**

In Alice Munro’s story “Boys and Girls,” a central theme is gender roles. Munro sets up a contrast between the roles boys/men and girls/women play in the story. The females are expected to do work…

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**MARRIAGE IS A PRIVATE AFFAIR**

**What are the Biblical allusions in “Marriage Is a Private Affair”?**

There are two Biblical allusions in Chinua Achebe’s short story “Marriage Is a Private Affair.” An allusion is a reference to a statement, person, place or event that is known from literature,…

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**THE CRUCIBLE**

**In The Crucible, what does John Proctor mean when he says, “Spite only keeps me silent. It is…**

On the morning that John Proctor is facing hanging, his sentence for being convicted of witchcraft, he and Elizabeth have a few minutes to speak before he is taken away. He is considering…

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**THE ABSOLUTELY TRUE DIARY OF A PART-TIME INDIAN**

**What are some ideas for haikus about Junior, Mary Runs Away, Mr. P, and Junior’s parents, using…**

The fifth chapter of The Absolutely True Diary of a Part-Time Indian is titled “Hope Against Hope,” and in this chapter, Mr. P tells Junior that he needs to leave the reservation if he wants to…

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**THE RAVEN**

**What is the meaning of the very last stanza of “The Raven”?**

The importance of the last stanza is made evident by the tense shift. The action changes from verbs like “Swung,” “tossed” and “shrieked” to “is sitting.” This decent into chaos for the poem’s…

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**TO KILL A MOCKINGBIRD**

**What is the setting of chapter two in To Kill a Mockingbird?**

The setting for Chapter 2, as for the rest of the novel, is Maycomb, Alabama during the 1930s. In terms of the specific timing of chapter two and its impact on the narration, the setting of chapter…

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**BLUES AIN’T NO MOCKINGBIRD**

**What does it mean when the narrator says the following in Toni Cade Bambara’s “Blues Ain’t No…**

In Toni Cade Bambara’s “Blues Ain’t No Mockin Bird,” immediately after Granny declines giving the men permission to film her home, the four children in her yard look at her expectantly because…

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**LITERATURE**

**Describe Ruth and Walter’s marriage in A Raisin in the Sun.**

Ruth and Walter’s marriage in A Raisin in the Sun is very strained for many reasons. One reason their marriage seems to be in trouble is that Ruth and Walter live in a tiny apartment with…

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**ANTIGONE**

**In Antigone, how do Antigone and Creon show commitment and suffering?**

In Antigone, Antigone and Creon both show commitment to the values they deem important. For example, Antigone is committed to living an honorable life and part of that commitment means living by…

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**A ROSE FOR EMILY**

**What are conflicts in “A Rose for Emily”? Are there any internal conflicts (conflicts within a…**

If by conflict you are referring to the traditional “x versus y” then there are several conflicts going on with Emily. The “versus” is the opposing force. Emily vs. her father- Old man Grierson,…

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**CARL SANDBURG**

**What are the similarities between the cat and the fog in “Fog” by Carl Sandburg?**

Since the poem is so short, I’ll post it here so it is easier to discuss. Fog The fog comes on little cat feet. It sits looking over harbor and city on silent haunches and then moves on. In this…

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**THE POSTMASTER**

**In what way does the postmaster feel out of place in the remote village?**

The postmaster feels out of place in the remote village’s primitive environment, and he feels decidedly uncomfortable in his interactions with the locals. In the story, we are told that the…

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**THE DOLL’S HOUSE**

**What are the two social class descriptions in “The Doll’s House”?**

The two social class descriptions that are present in Katherine Mansfield’s short story “The Doll’s House” are the upper class, which is represented by the Burnell sisters, and the very poor, lower…

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**OZYMANDIAS**

**What conversation occurs between the narrator and the traveler in the poem “Ozymandias?”**

In Percy Bysshe Shelley’s sonnet “Ozymandias,” two characters meet, and we might assume there has been a conversation. However, the narrator, the “I” of line 1, actually never speaks to the…

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**THE SCARLET IBIS**

**Explain what the ibis symbolizes in the story.**

In the short story, “The Scarlet Ibis,” the ibis is a beautiful bird found far from his home when a violent storm carries it away from its normal environment. Doodle and his family find the…

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**OF MICE AND MEN**

**Crooks, Candy, Lennie, and Curley’s wife are lonely people with specific needs. How can I compare…**

This is a great question. Crooks, Candy, Lennie, and Curley’s wife are lonely. More specifically, there is something about them that removes them from society. Crooks is the only black man on the…

2 educator answers

**THE TWENTY-ONE BALLOONS**

**The Higgins Balloon Factory was awarded the contract for decorating the avenue from the railroad…**

The Twenty-One Balloons is one of my favorite books, and I love picturing the scene in which the Higgons Balloon Factory is foiled in its efforts to complete its contract. This factory, which was…

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**JOHN DONNE**

**What is a translation of the poem “Air and Angels” by John Donne into modern day English?**

Air and AngelsBy John Donne Twice or thrice had I lov’d thee,I had already fallen in love with you two or three times, Before I knew thy face or name;Even before I had met you or learned your name….

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**I AM LEGEND**

**What is the setting for the book I Am Legend?**

Richard Matheson’s I am Legend has been made into several movies, including the recent Will Smith movie, and The Last Man on Earth, starring Vincent Price. The book itself is set in Los Angeles,…

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**OF MICE AND MEN**

**What are some examples of alienation in ”Of Mice And Men”?**

Alienation in Of Mice and Men is everywhere. As soon as Lennie and George get to the ranch, Slim remarks that he has not seen too many people travel together. This makes George and Lennie’s…

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**GUNS, GERMS, AND STEEL**

**Why was Oceania a good place to undertake the natural experiment in Guns, Germs, and Steel? I…**

It is not surprising that you cannot find “Oceania” in Guns, Germs, and Steel.” A quick scan of my digital copy shows that the word is used only in the notes. What is going on is that…

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**THINGS FALL APART**

**What is Okonkwo’s attitude towards the changes brought by the white Christian missionaries, and…**

Okonkwo reacts to the pervasive influence of the white colonizers the same way that he reacts to most threatening situations: he calls for violence and bloodshed to rectify the situation and…

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**MATH**

f(x)=x−2tan−1(x)f(x)=x-2tan-1(x) differentiating, f'(x)=1−2×2+1f′(x)=1-2×2+1 f'(x)=x2+1−2×2+1f′(x)=x2+1-2×2+1 f'(x)=x2−1×2+1f′(x)=x2-1×2+1 Now to find the absolute extrema of the function , that is continuous on a closed…

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**MATH**

The function is defined because x^2+x+1 always >0. Further, ln(y) is monotone increasing and therefore it has min and max where y has min and max. x^2+x+1 has a global minimum at x=-1/2 and it…

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**MATH**

**f(x)=x−ln(x),[(12),2]f(x)=x-ln(x),[(12),2] Find the absolute maximum and minimum values of f on the…**

This function is continuous on the given interval and is differentiable inside it. So it reaches minimum and maximum either at endpoint or where f'(x)=0. f(1/2) = 1/2-ln(1/2)=1/2 + ln(2)….

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**MATH**

**f(x)=xe−x28,[−1,4]f(x)=xe-x28,[-1,4] Find the absolute maximum and minimum values of f on the…**

f(x)=xe−x28f(x)=xe-x28 differentiating by applying product rule, f'(x)=x(e−x28)(−2×8)+e−x28f′(x)=x(e-x28)(-2×8)+e-x28 f'(x)=e−x28(−x24+1)f′(x)=e-x28(-x24+1)f'(x)=−14e−x28(x2−4)f′(x)=-14e-x28(x2-4) Now to find the absolute extrema of…

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**MATH**

**f(t)=t+cot(t2),[π47π4]f(t)=t+cot(t2),[π47π4] Find the absolute maximum and minimum values…**

f(t)=t+cot(t2)f(t)=t+cot(t2) differentiating, f'(t)=1−12csc2(t2)f′(t)=1-12csc2(t2) Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers that are…

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**MATH**

f(t)=2cos(t)+sin(2t)f(t)=2cos(t)+sin(2t) differentiating, f'(t)=−2sint+2cos(2t)f′(t)=-2sint+2cos(2t) Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers…

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**MATH**

**f(t)=8−t−−−−√3,[0,8]f(t)=8-t3,[0,8] Find the absolute maximum and minimum values of f on the…**

When t increases from 0 to 8, 8-t decreases from 8 to 0. And cubic root of 8-t also decreases monotonically from 2 to 0. Therefore the answer is: absolute minimum is 0 and absolute maximum is 2.

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**MATH**

**f(t)=t4−t2−−−−−√,[−1,2]f(t)=t4-t2,[-1,2] Find the absolute maximum and minimum values of f on the…**

f(t)=t4−t2−−−−−√f(t)=t4-t2 differentiating by applying product rule, f'(t)=t(12)(4−t2)−12(−2t)+4−t2−−−−−√f′(t)=t(12)(4-t2)-12(-2t)+4-t2 f'(t)=−t24−t2−−−−−√+4−t2−−−−−√f′(t)=-t24-t2+4-t2 f'(t)=−t2+4−t24−t2−−−−−√f′(t)=-t2+4-t24-t2…

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**MATH**

**f(x)=xx2−x+1,[0,3]f(x)=xx2-x+1,[0,3] Find the absolute maximum and minimum values of f on the…**

f(x)=xx2−x+1f(x)=xx2-x+1 differentiating by applying quotient rule, f'(x)=(x2−x+1)−x(2x−1)(x2−x+1)2f′(x)=(x2-x+1)-x(2x-1)(x2-x+1)2 f'(x)=x2−x+1−2×2+x(x2−x+1)2f′(x)=x2-x+1-2×2+x(x2-x+1)2 f'(x)=−x2+1(x2−x+1)2f′(x)=-x2+1(x2-x+1)2 Now to find the absolute…

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**MATH**

**f(x)=x+1x,[0.2,4]f(x)=x+1x,[0.2,4] Find the absolute maximum and minimum values of f on the given…**

Given: f(x)=x+(1x),[0.2,4]f(x)=x+(1x),[0.2,4] Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s). f'(x)=1−(1×2)=0f′(x)=1-(1×2)=0 1=(1×2)1=(1×2) x2=1×2=1…

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**MATH**

**f(x)=(x2−1)3,[−1,2]f(x)=(x2-1)3,[-1,2] Find the absolute maximum and minimum values of f on the…**

f(x) is continuous and differentiable everywhere, so it reaches its maximum and minimum either at the endpoint or where f'(x)=0. f(-1) = 0, f(2) = 27. f'(x)=3(x2−1)2⋅2xf′(x)=3(x2-1)2⋅2x . This is zero at x=0,…

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**MATH**

f(x)=3×4−4×3−12×2+1f(x)=3×4-4×3-12×2+1 differentiating, f'(x)=12×3−12×2−24xf′(x)=12×3-12×2-24x f'(x)=12x(x2−x−2)f′(x)=12x(x2-x-2) Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find…

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**MATH**

**f(x)=x3−6×2+5,[−3,5]f(x)=x3-6×2+5,[-3,5] Find the absolute maximum and minimum values of f on the…**

f(x)=x3−6×2+5f(x)=x3-6×2+5 differentiating. f'(x)=3×2−12xf′(x)=3×2-12x f'(x)=3x(x−4)f′(x)=3x(x-4) Now to find the absolute extrema of the function , that is continuous on a closed interval, we have to find the critical numbers…

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**MATH**

This function is differentiable everywhere. Therefore it reaches its absolute minimum and maximum either at the endpoints or where its derivative is zero. f(-2) = -16 – 12 + 24 + 1 = -3, f(3) = 54…

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**MATH**

**f(x)=5+54x−2×3,[0,4]f(x)=5+54x-2×3,[0,4] Find the absolute maximum and minimum values of f on the…**

Given: f(x)=5+54x−2×3,[0,4]f(x)=5+54x-2×3,[0,4] Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s). f'(x)=54−6×2=0f′(x)=54-6×2=0 54=6×254=6×2 9=x29=x2 x=3,x=−3x=3,x=-3 The…

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**MATH**

**f(x)=12+4x−x2,[0,5]f(x)=12+4x-x2,[0,5] Find the absolute maximum and minimum values of f on the…**

Given: f(x)=12+4x−x2,[0,5]f(x)=12+4x-x2,[0,5] Find the critical value(s) of the function by setting the first derivative equal to zero and solving for the x value(s). f'(x)=4−2x=0f′(x)=4-2x=0 4=2×4=2x 2=x2=x The value x=2…

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**MATH**

**f(x)=x−2ln(x)f(x)=x-2ln(x) Find the critical numbers of the function**

You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation f'(x) = 0. You need to evaluate the first derivative, using the product rule:

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**MATH**

**f(x)=x2e−3xf(x)=x2e-3x Find the critical numbers of the function**

You need to evaluate the critical numbers of the function, hence, you need to evaluate the soutions to the first derivative, such that: f'(x)=0f′(x)=0 You need to find the first derivative using the…

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**MATH**

**h(t)=3t−arcsin(t)h(t)=3t-arcsin(t) Find the critical numbers of the function**

You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation h'(t) = 0. You need to evaluate the first derivative: h'(t)=3−11−t2−−−−−√h′(t)=3-11-t2…

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**MATH**

**f(θ)=2cos(θ)+sin2(θ)f(θ)=2cos(θ)+sin2(θ) Find the critical numbers of the function**

You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation f'(θ)=0f′(θ)=0 . You need to evaluate the first derivative:

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**MATH**

**g(θ)=4θ−tan(θ)g(θ)=4θ-tan(θ) Find the critical numbers of the function**

You need to find the critical points of the function, hence, you need to evaluate the solutions to the equation g'(θ)=0g′(θ)=0 . You need to evaluate the first derivative:

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**MATH**

**F(x)=x45(x−4)2F(x)=x45(x-4)2 Find the critical numbers of the function**

Given: F(x)=x45(x−4)2F(x)=x45(x-4)2 Find the critical number(s) by setting the first derivative equal to zero and solve for the x value(s). F'(x)=x45[2(x−4)]+(x−4)2[(45)x−15]=0F′(x)=x45[2(x-4)]+(x-4)2[(45)x-15]=0…

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