Let [tex]\displaystyle f(x)= \lim_{n\to \infty}\left( \dfrac{n^n (x+n)\left(x+\dfrac{n}{2}\right) \dots \left(x+\dfrac{n}{n}\right)}{n! (x^2+n^2)\left(x^2+\dfrac{n^2}{4}\right)\dots \left( x^2+\dfrac{n^2}{n^2}\right)}\right)^{\dfrac{x}{n}}[/tex] for all x > 0 , then;

[tex] A ) \ f\bigg(\dfrac{1}{2}\bigg) \geq f(1) \\\\ B)\ f\bigg(\dfrac{1}{3}\bigg) \leq f\bigg(\dfrac{2}{3}\bigg) \\\\ C) \ f'(2)\leq 0 \\\\ D) \ \dfrac{f'3(3)}{f(3)}\geq \dfrac{f'(2)}{f(2)} [/tex]​

Answers

Answer 1

The solution is Option C.

The equation f'( 2 ) ≤ 0 is true

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

[tex]f ( x ) = \lim_{n \to \infty} [ \frac{( n^{n} ( x + n ) ( x + \frac{n}{2} ) ....( x + \frac{n}{n} )}{n! (x^{2}+n^{2}) ( x^{2} + \frac{n^{2}}{4} ) ....( x^{2} +\frac{n^{2}}{n^{2} } ) } ]^{\frac{x}{n} }[/tex]

Taking logarithm on both sides of the equation , we get

[tex]ln f ( x ) = \lim_{n \to \infty} \frac{x}{n} [\sum _{r=1}^nln\frac{1}{r/n} + \sum _{r=1}^nln\frac{(x + \frac{1}{r/n} )}{(x^{2} + (\frac{1}{r/n})^{2} ) }[/tex]

On simplifying the derivatives of the equation , we get

[tex]ln f ( x ) = x [\int\limits^1_0 {ln ( 1/t )} \, dt + \int\limits^1_0 {ln \frac{( x+1/t )}{(x^{2} + 1/t^{2} ) } } \, dt][/tex]

Now , substituting the limits in the equation , we get

[tex]ln f ( x ) = x [ \int\limits^1_0 {( - ln t+ ln (tx+1)-lnt-ln(t^{2} x^{2}+1)+lnt^{2}] } \, dt[/tex]

Substitute the value of tx = z in the equation , we get

[tex]ln f ( x ) = \int\limits^1_0 {ln\frac{(1+z)}{1+z^{2} } } \, dz[/tex]

Taking exponential on both sides of the equation , we get

[tex]f ( x ) = e^{\int\limits^x_0 {{ln\frac{(1+z)}{1+z^{2} } } \, dz} \, }[/tex]

Differentiating with respect to x , we get

[tex]\frac{f' (x ) }{f ( x ) } = ln\frac{1+x}{1+x^{2} }[/tex]

Now , on further simplification of the equation , we get

f ( x ) > 0 for x > 0

And , f' ( x ) > 0

[tex]ln\frac{1+x}{1+x^{2} } > 0[/tex]

And ,

x > x²

0 < x < 1

Therefore , f ( x ) is increasing ∀ x ∈ ( 0 , 1 )

Hence , the equation f' ( 2 ) ≤ 0 is true

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Answer 2

(A) False(B) True(C)True(D) True

Explanation:

[tex] \rm f(x) = \lim \limits_{n \to \infty } \left( \dfrac{ \displaystyle \rm \prod \limits_{r = 1}^n \left(1 + \dfrac{rx}n \right ) }{\displaystyle \rm\prod \limits_{r = 1}^n \left(1 + \bigg(\dfrac{rx}n \bigg)^{2}\right )} \right)^{ \dfrac{x}{n} }[/tex]

[tex] \large \rm = {e}^{ x \int \limits_{0}^{1} ln(1 + xy)dy - ln(1 + (xy {)}^{2} ) dy }[/tex]

[tex]\rm f'(x)=f(x) \ln \bigg( \dfrac{1 + x}{1 + {x}^{2} } \bigg)[/tex]

[tex]\rm For \: x \in(0,1) \: it \: i s \: increasing \: function[/tex]

[tex]\rm f'(2) = f(2) \ln( \frac{3}{5} ) < 0[/tex]

[tex]\rm \dfrac{f'(3)}{f(3)} =ln \bigg( \dfrac{2}{5} \bigg), \dfrac{f'(2)}{f(2)} = ln \bigg( \dfrac{3}5 \bigg) [/tex]


Related Questions

What's the common difference of the sequence -5, -2, 1, 4, 7, ...?
A) d= -5
B) d= 2
C) d=-3
D) d= 3

Answers

Answer:

The answer is D) d=3!!!!!

find the value of each variable

Answers

Answer:

x = 16  y= 16√3

Step-by-step explanation:

we know that a angle is 60 degrees. Thanks to this information we can say that x is half the hypothenuse, so is value is 16.

y can be express as √3/2 x 32 = 16√3

X^2(x+5)
Help me PLS

Answers

Answer: x^3+5x^2

Step-by-step explanation:

Joan paid $47.98 for a jacket. If the tax rate is 8.25%, what was the cost of the jacket before tax?

Answers

Answer:

$44.021

Step-by-step explanation:

$47.98 - 8.25% = $44.021

HELPPPPPPP PLZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

Answers

Answer:

answer is 9

Step-by-step explanation:

1st method: Proportion

15/100 = n/60

cross-multiply to get:

100n = 900

n = 900/100

n = 9

2nd method: Convert % to Decimal and Multiply

.15 × 60 = 9

Sketch the graph of the following equations: y=x+5, y=-(x+5), and y = |x+5|

Answers

is a great game but I can’t have any problems at the same level that you have a few things that 234x234

The graph of functions y=x+5, y=-(x+5), and y = |x+5| are shown in figure.

What is Equation of line?

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

Equations are,

y = x + 5

y = - (x + 5)

y = |x + 5|

Since, All equations shows equation of line.

Hence, We can draw all the equations as shown in figure.

y = x + 5

Slope = 1

y - intercept = 5

y = - x - 5

Slope = - 1

y - intercept = - 5

Thus, The graph of functions y=x+5, y=-(x+5), and y = |x+5| are shown in figure.

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i need help with this

Answers

Answer:

<FNM = 73

Step-by-step explanation:

we are given

GNF = 60

MNL = 47

60+47=107

107-180=73

Steve reaches into a bag with 5 marbles 3 yellow marbles and 2 green marbles and randomly selects one the he puts the marble back and sam reaches into the same bag and randomly selects a marble is this independent or dependent

Answers

Answer:

Independent

Step-by-step explanation:

The reliance of an event on the outcine of another event gives rise to a dependent event. For any two or more event, whereby the outcome of event A has no effect on the possible outcome of event B, then such scenario is termed as being Independent. In the scenario described above, the outcome oof Sam's pick does not depend on the outcine of Steve's pick. This is because, once Steve makes his selection, the marble is returned into the box before Sam makes his own selection.Therefore, no matter the color of marble picked by Steven, it will have no bearing on the outcome of Sam's pick.

Can someone help me with this one too please

Answers

Answer:

1) f - 2

2) 45 + b

3) s - 82

4) 52c

5) t/5

Answer:

1) f-2

2) 45+b

3) s-82

4)52c

5)t/5

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HELP I NEED HELP ASAP

Answers

The answer i think is it c
I believe the answer is C

Find the measure of the sixth interior angle of a hexagon if the measure
of the other five interior angles are 80°, 90°, 130°, 130°, and 150°. (Show
all work)

Answers

Answer:

140°

Step-by-step explanation:

The sum of interior angles of a n-agon is (n - 2) * 180°

a hexagon should have the sum of angles equal to

4 * 180° = 720°

Now we know that the other five angles have total measure

80° + 90° + 130° + 130° + 150° = 580°

So the sixth one has to be:

720° - 580° = 140°

Answer:140°

Step-by-step explanation: required equation : 180*(n-2)

since it is a hexagon, the sum of all interior angles=180*(6-2)

                                                                                       =180*4 =720°

now the measure of 6th angle=720 -(80+90+130+130+150)

                                                    = 720 -580

                                                   = 140

Priscilla received 240 votes in the election for student council president. That was 60% of the students who voted in the election. How many students voted in the election? *

Answers

Answer:

400 students voted in the election

Step-by-step explanation:

Given:

Total number of votes received by Priscilla = 240

This was 60% of the students who voted in the election.

To find: Total number of students who voted in the election

Solution:

Total number of votes received by Priscilla = 60% of Total number of students who voted in the election

240 = 60% of Total number of students who voted in the election

240 = [tex]\frac{60}{100}[/tex] × Total number of students who voted in the election

So,

Total number of students who voted in the election [tex]=\frac{240(100)}{60}=400[/tex]

It wants me to find m∠2 from this

Answers

Answer:

Alternate interior angles have the same measure. As we know that 'angle 1 ' and 50° are alternate interior angles hence they must be equal. Also we now that sum of all the angles of a triangle is equal to 180°. Hence, m∠2=15°

Step-by-step explanation:

Answer:

90°

Step-by-step explanation:

This is a kite.

The angle measures of the diagonals (the lines in the middle) are always going to be 90° if they looks like this without any other measures. This goes for all other quadrilaterals as well.

m∠2=90°

Hope I helped.

Troy and two of his friends help troy's dad do yard work. After his dad paid $35 for gas, the boys made 128. How much did each child make?

Answers

The boys made about 64$ each.

PLEAS EHELP ILL GIVE BRAINLIEST

PLEASE SHOW ALL WORK AND BE SPECIFIC

Answers

Answer:

Where is the question? You didn't post it.

What is the sum of (3x – 6) + (2x + 5)?

Answers

Answer: 5x - 1

Step-by-step explanation: 3x + 2x = 5x and -6 + 5 = -1, so 5x - 1.

Answer:

The sum of (3x – 6) + (2x + 5) is 5x - 1

Step-by-step explanation:

(3x - 6) + (2x + 5)

3x - 6 + 2x + 5

3x + 2x - 6 + 5

5x - 1

Thus, The sum of (3x – 6) + (2x + 5) is 5x - 1

-TheUnknownScientist

PLEASE HELP! this is a 4 part problem but here is the last part :)!!!

Answers

The answer is B 63ft squared

A student incorrectly says the volume of the regular hexagonal​ prism, to the nearest cubic​ centimeter, is 1,619 cm3. What is the correct​ volume? What error did the student most likely​ make?
The volume of the regular hexagonal prism is ___ cm3.

Answers

Answer:

17.5

Step-by-step explanation:

Given: BD bisects AE, A = E
Prove: ABC EDC
(No links)

Answers

Answer:

All sides are equal better explenation bellow

Step-by-step explanation:

It's a mirror flip of a ABC triangle

Yet you can say AB Is = to DE

And since most likely it's a 60° triangle all the angles are 60°

All triangles have 180°

Which means AB =BC

BC= AC

BC =CD

DE= CD

CE=DE

Which means all the sides are equal

Ashleys Shoe Store buys shoes for $35 and then sells them for
$85.
What is the percent of increase on the price of the shoes?
Round to the nearest whole number

Answers

85/35 = 2.43% because that is the percent you would need to multiple by

Area of Composite Figures
-1.5 in.
10 in.
A
6 in
4 in.
1.5 in.
17 inches squared
0.40 inches squared
52 inches squared
0 61 inches squared

Answers

Answer:

Step-by-step explanation:

The answer is 61in that’s the answer to your problem

COMPOSITE FIGURES!!! can you please help​

Answers

I believe it would be 470 cm.

For the blue figure, there are two 2 by 2 panels and three 7 by 2 panels that would be exposed to the surface

So (7•2)•3 = 42 and (2•2)•2 = 8 so 42•8=50

The green panel is made up of four 12 by 7 panels and two 7 by 7 panels, with one of them being covered with a 14 cm part.

So (7•12)•4 = 336 and 7•7 = 49. The top part would be 49-14=35 so 336 + 49 + 35 = 420

And 420 + 50 = 470

Please correct me if I am wrong, have a good day!

What is the surface area of the rectangular prism that can be formed using this net?

Answers

Answer:

S = 2(lw + lh + wh)Step-by-step explanation:

What value of x will make the equation below true?
14−(x+7)=3

Answers

14-x-7=3
7-x=3
-x=-4
x=4

2/14 as a decimal (please explain)

Answers

Answer:

Hi! The answer to your question is 0.142857. (Convert the fraction to a decimal by dividing the numerator by the denominator)

Step-by-step explanation:

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☆Brainliest is greatly appreciated!☆

Hope this helps!!

- Brooklynn Deka

Calculate the area of the shaded area !!
EZ FREE POINTS!!

Answers

Answer:

20 sq units

Step-by-step explanation:

Answer:

20 sq units

Hope it helps :)

Can some one help? PLEASE!!!!!!!!

Answers

1: true
2: false
3: false
4: true
5: true

I hope these are correct!
true
false
false
true
true

:-)

Root5 /2 root5 -1
can be written in the form
a+rootb /c
where a, b and c are integers.
Find the values of a, b and c when the resulting fraction
is written in its simplest form.

Answers

Answer:

10, 5, 19

Step-by-step explanation:

Recall that

[tex] {a}^{2} - {b}^{2} = (a + b)(a - b)[/tex]

So we can multiply the fraction by

[tex] \frac{2 \sqrt{5} + 1 }{2 \sqrt{5} + 1 } [/tex]

And get

[tex] \frac{ \sqrt{5} }{2 \sqrt{5} - 1} \times \frac{2 \sqrt{5} + 1 }{2 \sqrt{5} + 1 } = \frac{10 + \sqrt{5} }{20 - 1} = \frac{10 + \sqrt{5} }{19} [/tex]

A scale drawing of a dance floor is shown. What is the area of the actual dance floor? 22 cmon the top and bottom and 14 cm on the sides 5 cm equals 8 feet

Answers

Answer:

The area of the dance floor will be 788.48ft²

Step-by-step explanation:

Since 5cm represents 8ft, it means

1cm represents 8/5ft.

The scale length is 22cm, therefore the actual length would be

22 x 8/5ft = 35.2ft

The scale breadth is 14cm, therefore the actual breadth would be

14 x 8/5ft = 22.4ft

The area of the dance floor will be

(35.2ft)(22.4ft) = 788.48ft²

what is the same as 8/9?​

Answers

Answer:

8×1/9

Step-by-step explanation:

It is same as 8/9 because 8×1=8

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