5

Find the absolute raximum and absolute minimum values of f on the given interval,Tx) =< +[0.2absolute minimum valueapsolute maxlmum value...

Question

Find the absolute raximum and absolute minimum values of f on the given interval,Tx) =< +[0.2absolute minimum valueapsolute maxlmum value

Find the absolute raximum and absolute minimum values of f on the given interval, Tx) =< + [0.2 absolute minimum value apsolute maxlmum value



Answers

Find the absolute maximum and absolute minimum values of $f$ on the given interval.
$f(x)=\frac{x}{x^{2}+1},[0,2]$

Hello. We have to find the absolute maximum and absolute minimum value of the function on the given in trouble. And the effect is given ex upon X squared minus X plus one. And the interval is zero comma three. Okay, so now the first we will find after sex. So it will be X squared minus X plus one. Mm. The defense is enough. Access burn minus of differences and denominator that is two X minus one multiplied by x upon X squared minus X plus one. Okay, so it will, because two X squared minus X plus Fun man is up to X plus one mhm upon X squared minus X plus one whole is good. So it will be X squared minus of three eggs plus two upon excess square minus X plus one whole square so we can find the critical point that it's have the sex by FDA Sex close to zero. So X squared minus three expressed to close to zero. So access got minus two x minus. X plus two equals to zero, So x minus two. Money is one X minus two because 20 from here we will get X minus one x managed to cause too zero Mhm. So from here we will get the critical number X equals to one and expect us to. So now we will find the value of the function at one had to and also so first people calculate this and we will get the value of F one. It will be f monikers to one and I have to close to two or three. And also we will calculate the value of the function. At the end point of the interval and intervals you go 13 so f of zero will be zero. And I have trouble with three by seven. So from here we can die. Absolute maximum at X equals to one. The value of the and value of fun function will be one and absolute minimum value had acceptance to zero. The value of half of zero will be zero. I hope you're in the short. Thank you.

But the thing you probably want to find the absolute maximum and absolute minimum values on the given interval. So we're going to consider ffx Equalling need to the negative acts And it's either the -2 x Very like from 0 to 1. So consider after zero one. But then we also want to consider critical points in finding the absolute maximum and minimum. So with that we're going to consider after crime effects and we're going to have f prime of X. Um we get a zero right here 0.693. So plugging that value in. Mhm. You see that this is going to be the absolute maximum at 0.25 And the absolute minimum is up zero.

Here. We're looking for the extreme values of this function. F of X equals X plus one over X on the closed interval from 0.224 So before I do anything, I want to find the domain of this function. Notice I haven't extends nominator here and I can't divide by zero. So therefore we have to exclude zero from the Joe Me. Now we're going to find the first derivative F prime of X. Before I do that, I'm going to rewrite the original function and just bring this ex up to the numerator. So I have X plus X of negative one instead of X plus one over X. I'm doing this so I can use the power rule to differentiate. So using the power rule, the derivative of X is just one and the derivative of exit minus one. I have to multiply down by this exponents and then subtract one from the exponents. So I get the derivative F prime of X equals one minus X to the minus two. I'm going to rewrite the derivative here just so it's easier for us to find the critical values. I'm going to bring this negative exponents down into the denominators, so it becomes a part of a positive exponents, and then we're gonna get a common denominator, so I could rewrite the derivative as one big fraction. So I'm going to multiply here by X squared over X squared, and that gives us one fraction X squared minus one all over X square. Now, to find the critical values, there are two things I need to check. The first is where the derivative equals zero. So since we have a fraction, the only time a fraction equals zero is when the numerator equal zero. So I'm just taking the numerator X squared minus one and setting that equal to zero. Solving this for X gives me two values one and negative one. The second thing I need to look for when we're solving for the critical values is where the derivative is undefined. Since I have a fraction, a fraction is undefined whenever the denominator is equal to zero. So I'm just taking the nominator here X squared setting that equals zero in solving for X. That gives us zero. But this is not a critical value because zero is not in our domain so the only two critical values that we have our one and negative one. Now we're looking for the largest and smallest value that this function takes on this interval. So I'm going to evaluate the function F that both of the endpoints of the interval and only one of these critical values because X equals negative one, is not inside this interval. So I'm plugging these values into the original function, not the derivative of the function. If I plug them into the derivative of the function, that would give me the slope of the function at those X values, not the actual function value. So plugging in 0.2 gives me 5.2 plugging and four gives me 4.25 and the only critical value that I care about is one. So I'm just plugging in one into the original function that gives me a value of two. Now, from here I'm looking for the biggest and the smallest. So the absolute maximum value here is 5.2, and that happens when X equals 0.2 and the smallest value here, the absolute minimum is going to be, too. And that happens when expose one

All right, let's go ahead and answer this question. We have the function. Why equals two X over X squared minus X plus one X is defined to be between zero and three inclusive. And it's asking you what's the what are the absolute extreme. Okay, so whatever you solve these kind of problems, there are two things you need to keep in mind. First, always check the endpoints. Okay, because the extreme value theory, um basically says that if there if a function has an extreme value, it will always happen at the end point or somewhere in between. Okay, So if the end points are included in the in the analysis, we always check, So we're going to plug in zero and three to figure out whether it's going to be an extreme. Okay, The second thing you want to keep in mind is to check the critical points. So we're going to look at the derivative, let it equal to zero. And if you want to be very precise, we do want to check the second derivative to see if it's going to be a local maximum or local minima or by observing the first derivative and check if it's actually going to change the signs. But I am going to admit, omit that detailed process of that. Okay, so I'm going to assume that when the derivative is equal to zero, it actually is a critical point rather than a saddle. Okay, so let's go through the calculation here. First, I am going to call this F and plug in. Zero f of zero is very simple. The numerator is zero. The denominator is not zero. So it'll end up being zero next f off three. A quick calculation tells you that the top is three. Bottom is nine minus 366 plus one is seven. So it's 37 So at least I know that zero is not an absolute maximum seven third probably. Okay, Now let's check F prime. This is a question so we can use the quotient rule. So it is going to be ex prime times the denominator. Of course, Ex prime is one. So you will have X squared minus X plus one minus the denominator prime. So it is going to be two x minus one times the numerator, which is X now the denominator. Of course you're gonna have to square it, so I'm just gonna But I'm just gonna call it triangle. And the reason is I know that I'm gonna let this equal to zero. And if you observe that the denominator is an irreducible quadratic meaning that the B squared minus four A. C is a positive number, uh, negative number. The denominator is never going to be negative. So I don't have to worry about the case where there's a division by zero or anything like that's all. I'm gonna omit it because the numerator simplifies to one minus X squared. And if I multiply the denominator on both sides, it disappears anyway, so I don't have to worry about that. Okay, so you can see that X is plus or minus one. But one is the only number that belongs in this limit. So we know that one is the only point that we need to check. Let's figure out what effort of one is equal to the gnome writers one and the denominator X squared minus one is simply zero. So the remaining one makes this equal to one. And one is, of course, larger than 3/7. Okay, so from this analysis, we figured out that this guy, it is going to be the local minimum or the absolute minimum. Excuse me. The bottom one here is going to be a an absolute maximum. And 7 30 is just a point at the end point. If you use a graphing utility, you will actually see that this is a type of graph that looks something like this and 0 to 3 is actually a region from here. So there. And this is going to be one. This is a zero, and this height right here is the seven third that we actually observed. And this is how I answer this problem.


Similar Solved Questions

5 answers
Let A be an n * n matrix. Show that the matrix A is invertible if and only the column vectors of A span R"
Let A be an n * n matrix. Show that the matrix A is invertible if and only the column vectors of A span R"...
5 answers
Find the sum of the following scrics. 13) 2 N2 k#014) 2 e-9k klio
Find the sum of the following scrics. 13) 2 N2 k#0 14) 2 e-9k klio...
5 answers
If ln a =2 In b = 3,and In c = 5,evaluate the following: 02 (a) In( 6323 ,(b) lnVa364c4 = =Preview
If ln a =2 In b = 3,and In c = 5,evaluate the following: 02 (a) In( 6323 , (b) lnVa364c4 = = Preview...
5 answers
Fand Amides 313 AmincsAcetamideBenzamide2. Odor'uwuaissa. Lo F3 . SolubilityD_ Hydrolysis of an Amide Acid HydrolysisU 0 #ount /0a T Jeaf bizn %uodw? L ULo{ 4' 4 r"#eniaib E ~Eu"ecV 3nonu41nja- AcetamideLn , Alpub. Benzamide
Fand Amides 313 Amincs Acetamide Benzamide 2. Odor 'uwuaissa. Lo F 3 . Solubility D_ Hydrolysis of an Amide Acid Hydrolysis U 0 #ount /0a T Jeaf bizn %uodw? L ULo{ 4' 4 r"#eniaib E ~Eu"ecV 3nonu 41nja- Acetamide Ln , Alpu b. Benzamide...
5 answers
Solve Bernov Il; € quHn^8+y y
Solve Bernov Il; € quHn^ 8+y y...
5 answers
Enter your answer in the provided box_The equation for the metabolic breakdown 0f glucose (C6H,206) is the same as the equation for the combustion of glucose in air:CoH,zO6(s) 602(g) 6COz(g) 6HzO()Calculate the volume of COz produced at 37*C and 1.00 atm when 4.65 g of glucose is used up in the reaction_
Enter your answer in the provided box_ The equation for the metabolic breakdown 0f glucose (C6H,206) is the same as the equation for the combustion of glucose in air: CoH,zO6(s) 602(g) 6COz(g) 6HzO() Calculate the volume of COz produced at 37*C and 1.00 atm when 4.65 g of glucose is used up in the r...
5 answers
Electrc power the rate which electrical enriy produced and" measureo Watts (W): The tota amount TBY produced over given time period Measured Watt - hours (Wh): Ona centaino3v, solar panel produces energy at a rate Watts given by the function P(t), where t is the numbe hours since sunrise 6-C0 AM: (12 pts) Interpret the following practica terms, including units. Use complete Rentences P(2) = 125P'(2S) = 52(198)J3 P(t) dt 477
Electrc power the rate which electrical enriy produced and" measureo Watts (W): The tota amount TBY produced over given time period Measured Watt - hours (Wh): Ona centaino3v, solar panel produces energy at a rate Watts given by the function P(t), where t is the numbe hours since sunrise 6-C0 A...
1 answers
Evaluate the integral. $$ \int_{1}^{2} \frac{x}{2 x^{2}-1} d x $$
Evaluate the integral. $$ \int_{1}^{2} \frac{x}{2 x^{2}-1} d x $$...
5 answers
Cakculate the % m/m of _ solution made by dissolving 50.0 of D-glucose 950.0 mL of water (d 1.00 g/mL)
Cakculate the % m/m of _ solution made by dissolving 50.0 of D-glucose 950.0 mL of water (d 1.00 g/mL)...
5 answers
41-54 Equations Involving Fractional Expressions The givenequation is either linear or equivalent to a linear equation. Solvethe equation.$$ rac{3}{x+1}- rac{1}{2}= rac{1}{3 x+3}$$
41-54 Equations Involving Fractional Expressions The given equation is either linear or equivalent to a linear equation. Solve the equation. $$\frac{3}{x+1}-\frac{1}{2}=\frac{1}{3 x+3}$$...
5 answers
Suppose the amount substancecerain rdiouctiSoinoideCvs Ing T70 mg 1,30 mg Ovc; period of 20 2 vears, Calculate thehalcpl theRound your answer to significant digits_year
Suppose the amount substance cerain rdioucti Soinoi deCvs Ing T 70 mg 1,30 mg Ovc; period of 20 2 vears, Calculate thehalc pl the Round your answer to significant digits_ year...
5 answers
Question 13Domain archaea is best described as a prokaryote:TrueFalse
Question 13 Domain archaea is best described as a prokaryote: True False...
5 answers
Fcr = semple ofn = 15 individuals; how large Peatson contc Iation necessary CS?be statistically signiticant for & two-lailed tcst with a =Select one:05140,4783.3290,483
Fcr = semple ofn = 15 individuals; how large Peatson contc Iation necessary CS? be statistically signiticant for & two-lailed tcst with a = Select one: 0514 0,478 3.329 0,483...
5 answers
QUESTION 5: (10 marks)a) What is vaccine? In your opinion, why vaccination program is important? (2 mark)b) Describes vaccines that were developed by Pasteur and Chamberland_(4 marks)c) What is quorum sensing and explain why it is important for bacteria?marks)
QUESTION 5: (10 marks) a) What is vaccine? In your opinion, why vaccination program is important? (2 mark) b) Describes vaccines that were developed by Pasteur and Chamberland_ (4 marks) c) What is quorum sensing and explain why it is important for bacteria? marks)...

-- 0.020514--