A 2-column table with 5 rows. Column 1 is labeled Number of Apples with entries 0, 1, 2, 3, 4. Column 2 is labeled Number of Slices with entries 0, 6, 12, 18, 24.
Use the table to find the constant of proportionality and the equation that represents the relationship.

Let x = number of apples and y = number of slices.

The constant of proportionality is
.
The equation for this relationship is

Answers

Answer 1

Answer:

The constant of proportionality is 6. This makes the relation y = 6x.

Step-by-step explanation:

Concept: As the number of apples are increasing, the number of slices are also increasing. Given that x is the number of apples and y is the number of slices. This means that y is directly proportional to x.

y ∝ x

Let the constant of proportionality be k.

y = kx

For x = 1, y = 6. So, k = 6.

For x = 2, y = 12. So, k = 6.

This follows for all the values.

Clearly, the constant of proportionality is 6.

The relationship will become y = 6x.

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Related Questions

what is the answer for this question

Answers

Answer:
288 minutes.(4.8 hours)
Step-by-Step-Explanation:
3 people can build a shed in 8 hours, so how long would it take 1 person to?
3 / 3 = 1
8 * 3 = 24

It would take one person 24 hours to build the shed alone.
24 hours to minutes -
24 * 60 = 1440

If 5 people were to build a shed
(1440 minutes / 5 people -(1440/5= 288))

It would take 5 people 288 minutes (4.8 hours) to build a shed

The weight of a cat is normally distributed with a mean of 9 pounds and a standard deviation of 2 pounds. Using the empirical rule, what is the probability that a cat will weigh less than 11 pounds?

Answers

If the value of the z-score is 1. Then the probability that a cat will weigh less than 11 pounds will be 0.84134.

What is the z-score?

The z-score is a statistical evaluation of a value's correlation to the mean of a collection of values, expressed in terms of standard deviation.

The z-score is given as

z = (x - μ) / σ

Where μ is the mean, σ is the standard deviation, and x is the sample.

The weight of a cat is normally distributed with a mean of 9 pounds and a standard deviation of 2 pounds.

Then the probability that a cat will weigh less than 11 pounds will be

The value of z-score will be

z = (11 – 9) / 2

z = 1

Then the probability will be

P(x < 11) = P(z < 1)

P(x < 11) = 0.84134

Thus, the probability that a cat will weigh less than 11 pounds will be 0.84134.

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Maite's rent increased by 6%. The increase was $97.8. What was the original amount of Maite's rent? Please show me how to solve it as well please

Answers

Answer:

1630

Step-by-step explanation:

In words you are looking for 6% of what number is 97.80, turn that into an Algebra equation .06x = 97.80 so x = 97.80/.06 so x = 1630


Select the correct answer.
Identify the end behavior and the zeros of function h.
h(1) = -1³ - 91² +41 +96
Based on these key features, which statement is true about the graph representing function h?
A.
The graph is negative on the intervals (- infinity, -8) and (-4, 3).
B. The graph is positive on the intervals (-8, -4) and (3, infinity).
OC. The graph is negative on the intervals (-3, 4) and (8, ∞).
D. The graph is positive on the intervals (-infinity ,-8) and (-4, 3).

Answers

Based on the key features, the end behavior and the zeros of function h [h(x) = -x³ - 9x² +4x +96], the statement that is true of the above graph is: "The graph is positive on the intervals (-8, -4) and (3, infinity) (Option B)"

What is end behavior?

A function's "end behavior" refers to how the function's graph behaves at its "ends" on the x-axis.

In other words, if we look at the right end of the x-axis (as x approaches + ∞) and the left end of the x-axis (as x approaches - ∞ ), the end behavior of a function represents the trend of the graph.

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I need answer as fast as possible please.​

Answers

Step-by-step explanation:

a=110. x,55

y=180-(x+75)=50

w,75

b,110

x,40

y,30

Solve for w.
−16w-3 = 5w²

Answers

Answer:

w = -1/5

OR

w = -3

Step-by-step explanation:

Given equation:

−16w-3 = 5w²

Solution:

Subtracting 5w^2 from both sides,we get

-16w-3-5w² = 5w² - 5w²-5w²-16w-3=0

Factor the LHS of this equation using middle term factor:

(-5w²-1)(w-3)

Now,

[tex]( - 5w - 1) = 0 \: \: \: \: \: \: \: \: ...(1)[/tex][tex](w - 3) = 0 \: \: \: \: \: \: \: \: \: \: ... (2)[/tex]

Solving for equation 1:

[tex] - 5w = 0 + 1[/tex][tex] - 5w = 1[/tex][tex] \boxed{w = - \cfrac{1}{5} }[/tex]

Solving for equation 2:

[tex]w - 3 = 0[/tex][tex]w = 0 - 3[/tex][tex] \boxed{w = - 3}[/tex]

[tex] - 16w - 3 = 5 {w}^{2} \\ \\ 0 = 5 {w}^{2} + 16w + 3 \\ \\ 5 {w}^{2} + 16w + 3 = 0 \\ \\ 5 {w}^{2} + w + 15w + 3 = 0 \\ \\ (5 {w}^{2} + w) + (15w + 3) = 0 \\ \\ w(5w + 1) + 3(5w + 1) = 0 \\ \\ (w + 3)(5w + 1) = 0. [/tex]

The value of w is -3 and -1/5 .

1 in = 2.54 cm
how many millimeters are in 10.5 feet?
A.266.7 mm
B. 1,260 mm
C. 320.04 mm
D. 3,200.4 mm

Answers

Answer:

[tex]\fbox {D. 3,200.4 mm}[/tex]

Step-by-step explanation:

Given :

[ 1 inch = 2.54 centimeters ]

Unit conversions to keep in mind :

1 feet = 12 inches1 cm = 10 mm

Solving

10.5 feet10.5 x 12 inches126 inches126 x 2.54 cm320.04 cm320.04 x 10 mm3200.4 mm
The answer is D 3,200.4 mm I get that answer be equal ovulating the millimeters by 10.5 feet divided by the equal force of us in typical jetstream. Therefore 1 inches 2.54 cm. Then if I divide that by two, I get 3,200.4.

Which of the following sets of ordered pairs represents a function?

{(-6,-1), (13,8), (1,6), (1,-10)}

{(10,5), (10,-5), (5,10), (5,-10)}

{(3,5), (-17,-5), (3,-5), (-17,5)}

{(10,5), (-10,-5), (5,10), (-5,-10)}

Answers

Answer:

Step-by-step explanation:

A function can only have one output for an input.  That is, for any value of x, there must be a unique value of y.

{(-6,-1), (13,8), (1,6), (1,-10)}   Not a  Function:  (1,6) and (1,-10)

{(10,5), (10,-5), (5,10), (5,-10)}  Not a  Function:  (10,5) and (10,-5)

{(3,5), (-17,-5), (3,-5), (-17,5)}   Not a Function:  (3,5) and (3,-5)

{(10,5), (-10,-5), (5,10), (-5,-10)}  Function:  No duplicate values of y for a value of x.

area of rectangle = l×b find area of rectangle in sq cm
a) l=7cm,b=4cm

Answers

Answer:

[tex]28{cm}^{2} [/tex]

Step-by-step explanation:

we know

area of rectangle=l*b=(7*4)sq cm=28sq cm

which of the following must be true?

Answers

Answer:

C

Step-by-step explanation:

Answer C is correct.  The absolute value of 10 is 10 and that of -10 is 10.  Same result.

L
N
(x-4) in. O
(x-3) in.
(x + 2) in.
x in.
K
Which value of x would make NO || KJ?
1
6
08
O 10

Answers

Answer:

x = 8

Step-by-step explanation:

[tex]\sf If\:\: \overline{NO} \parallel \overline{KJ}\:\:then\:\: \triangle LNO \sim\triangle LKJ[/tex]

Therefore:

[tex]\implies \sf \overline{LN} : \overline{LO} = \overline{LK} : \overline{LJ}[/tex]

[tex]\implies (x-3):(x-4)=(x-3)+(x+2):(x-4)+x[/tex]

[tex]\implies \dfrac{x-3}{x-4}=\dfrac{2x-1}{2x-4}[/tex]

[tex]\implies (x-3)(2x-4)=(2x-1)(x-4)[/tex]

[tex]\implies 2x^2-10x+12=2x^2-9x+4[/tex]

[tex]\implies -10x+12=-9x+4[/tex]

[tex]\implies 12=x+4[/tex]

[tex]\implies x=8[/tex]

Can anyone help me with this

Fnd the value of x.

x = ?

Answers

Answer:

X=62 degrees

Step-by-step explanation:

The solution is in the image

Answer:

62°

Step-by-step explanation:

We know that the sum of the interior angles in a triangle is added up to 180°.

Therefore,

68.5° + 49.5° + x = 180°

118° + x° = 180°

x = 180° - 118°

x = 62°

In the above diagram, the demand for pepper is an example of price __________.
A.
elasticity
B.
shifting
C.
inelasticity
D.
controlling


Please select the best answer from the choices provided

A
B
C
D

Answers

Answer:

B. shifting

Step-by-step explanation:

When the price is rising or shifting higher, the demand curve moves to the left. But, t here is a corresponding change in the demand curve in that scenario while the cost stays unchanged. Since it relies on variables other than price, the simultaneous shift might go either left or right. However, the upward-pointing line depicts a rise in price and a fall in supply. 

Angelina's family owns a mini-golf course. When discussing the business with a customer, she explains there is a relationship between the number of visitors and
hole-in-one winners. If x is the number of visitors and y is the number of winners, which conclusion is correct?
A. The ordered pair (-3, 6) is viable.
B. The ordered pair (7, 2) is viable.
C. The ordered pair (15,-7) is viable.
D. The ordered pair (18,3) in non viable

Answers

The ordered pair (7,2) is viable and Option B is the correct answer.

What is Relationship ?

Relationship between variables defines the way one variable is dependent upon the other variable.

It is given that x is the number of visitors and y is the number of winners,

It has to be seen and chosen that which ordered pair makes sense

The ordered pair is viable if the no. of visitor is positive and more than the number of winners.

Therefore ordered pair (7,2) is viable and Option B is the correct answer.

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Question 3 of 10
Which choice represents the simplified exponential expression?
(12-4)8
OA. 12-32
B. 12-12
O C. 12
OD. 124

Answers

The correct value that equates to this expression is 12‐³². Letter A

.

To solve this expression, just: eliminate the parentheses and multiply the exponents among themselves;[tex] \boxed{ \large \sf (a {}^{n} ) {}^{m} \rightarrow a {}^{n \times m} } \\ \\ [/tex]

Resolution

[tex]{ = \large \sf (12{}^{-4} ) {}^{8} } [/tex]

[tex]{ = \large \sf 12{}^{-4 \times 8} } [/tex]

[tex] \pink{ \boxed{ = \large \sf 12{}^{-32} } } \\ [/tex]

Therefore, the answer will be 12‐³²

Jose rides his bike for 5 minutes to travel 8 blocks he rides for 10 minutes to travel 16 blocks which value will complete the table

Answers

Using the unit rate, the missing values that completes the table are:

A = 5; B = 15; C = 40

How to Find Unit Rate?

Unit rate (m) = change in y/change in x.

5 minutes for 8 Blocks (5, 8) and 10 minutes for 16 blocks (10, 16)are given.

Unit rate (m) = (16 - 8)/(10 - 5) = 8/5

An equation that will define the function is, y = 8/5x. Use it to complete the table.

Find A (y) when x is 5:

y = 8/5(5) = 8

The value of A is: 5

Find B (x) when y is 24:

24 = 8/5(x)

5(24) = 8x

120 = 8x

120/8 = x

15 = x

The value of B is: 15

Find A (y) when x is 25:

y = 8/5(25) = 40

The value of C is: 40

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1. At which point do Line CF and Line GF intersect? They intersect at point?

2. Look at Line AD and Like BE. Do these lines intersect?
(a) yes they will intersect at Point F?
(b) no they will never intersect?
(c) yes they will point at Point G
(d) yes they will intersect at Point F?

3. Look at Line BG and Line AC. Where do they intersect? They intersect at Point?

(Please hurry giving 50 points!) ​

Answers

Answer: i think no?

Step-by-step explanation:

AD and BE are both parallel lines (they are parallel to eachother), so they will never intersect


CF and GF intersect at point F (i think)


Bg and Ac intersect at point B (i think)


i dont want to give a definite answer in the event im wrong bc I just learned this like a few weeks ago-

One number is six times another number. Determine the two numbers if the sum of their reciprocals is 7/24
.

Answers

Answer:

x=24, y=4

Step-by-step explanation:

x=6y

1/x+1/y=7/24,

1/6y+1/y=7/24

1/6y+6/6y=7/24

(1+6)/6y=7/24

7/6y=7/24, then

6y=24

y=24/6

y=4

x=6y=6*4=24

Estimate the solution to the following system of equations by graphing.
OA (-1,-1)
OB. (1,-1)
oc (1)
D.
3x + 5y = 14
61 - 4y = 9

Answers

An equation is formed of two equal expressions. The estimated solution of the two system of equations is at (5/2,4/3). Thus, the correct option is D.

What is an equation?

An equation is formed when two equal expressions are equated together with the help of an equal sign '='.


The solution of the system of equation is the point at which the two lines will intersect as shown below. Therefore, the solution will be,

Solution = (5/2, 4/3)

Hence, the estimated solution of the two system of equations is at (5/2,4/3). Thus, the correct option is D.

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Choose all of the following angles that cannot
be an interior angle in a regular polygon.
40° 45° 108° 132° 179°

Answers

Answer:

40 45 because the minimum internal angle is 60

What is the solution to -2|x − 1| = -4? A. x = 3 B. x = -1 or x = 3 C. x = 1 or x = 3 D. No solutions exist.

Answers

Answer:

B

Step-by-step explanation:

-2|x - 1| = -4

|x - 1| = 2

since we are dealing with a function that brings 2 values to the same result, the reverse function (needed to find the values of x that create the result y) has 2 branches :

(x - 1) = 2

and

(x - 1) = -2

x - 1 = 2

x = 3

x - 1 = -2

x = -1

therefore, B is the right answer.

Find the range of the given function y = 3x + 2 for the domain 4 and -4. ​

Answers

Answer:

Range: (-10 , 14)

Step-by-step explanation:

Given information:

Equation: y = 3x +2Domain: (-4 , 4)

Range: (x , y)?

Plug in domain of x = -4 and x = 4 into equation to find range.

f(-4) = 3 * -4 + 2 = -10

f(4) = 12 + 2 = 14

Range: (-10 , 14)

Find the equation of the line in slope-intercept form containing the points (6, -1) and (-3, 2).

Answers

[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]

[tex]\qquad \tt \rightarrow \: y= - \cfrac{x}{ 3} + 1 [/tex]

____________________________________

[tex] \large \tt Solution \: : [/tex]

Equation of line (two point form) :

[tex]\qquad \tt \rightarrow \: (y - y_1) = \cfrac{y _1- y_2}{ x_1 - x_2} (x - x_1)[/tex]

[tex]\qquad \tt \rightarrow \: (y - 2) = \cfrac{2 - ( - 1)}{ - 3 - 6} (x - ( - 3))[/tex]

[tex]\qquad \tt \rightarrow \: (y - 2) = \cfrac{2 + 1}{ - 9} (x + 3)[/tex]

[tex]\qquad \tt \rightarrow \: (y - 2) = - \cfrac{3}{ 9} (x + 3)[/tex]

[tex]\qquad \tt \rightarrow \: (y - 2) = - \cfrac{1}{ 3} (x + 3)[/tex]

[tex]\qquad \tt \rightarrow \: y - 2= - \cfrac{x}{ 3} - \cfrac{3}{3} [/tex]

[tex]\qquad \tt \rightarrow \: y = - \cfrac{x}{ 3} - 1 \cfrac{}{} + 2[/tex]

[tex]\qquad \tt \rightarrow \: y = - \cfrac{x}{ 3} + 1[/tex]

Answered by : ❝ AǫᴜᴀWɪᴢ ❞

what is the slope of the line that is perpendicular to the line 3y=-5x+21
a -5/3
b -3/5
c- 3/5
d- 5/3

Answers

Step-by-step explanation:

the slope is the factor of x in an equation

y = ax + b

we have here

3y = -5x + 21

to get to the general format above we need to divide everything by 3 :

y = -5/3 x + 7

so, we see, the slope is -5/3.

the perpendicular (angle of 90°) slope is the original slope turned upside-down and with flipped sign :

3/5

so, I guess the correct answer option is c.

but it is not clear what you wrote there, as there is a "-" sign somehow in all 4 answers.

Given the function f(x) below, evaluate 3f(-2) + f(1).
if z ≤-3
3z²-2z if -3 -2√2-1
if x > 0
7x-2
f(x) = 3x² - 2x
pls help

Answers

The value of the expression 3f(-2) + f(1) is 45

Piecewise functions

Piecewise functions are functions that has two or more equations. They can consists of parabola and straight line.

From the equations, the point where x is -2 is f(x) = 3x^2 - 2x

f(-2) = 3(-2)^2 - 2(-2)
f(-2) = 12 + 4

f(-2) = 16

Similarly the point where x = 1 is -2√x - 1

f(1) = -2√1 - 1

f(1) = -2 - 1

f(1) = -3

Substitute

3f(-2) + f(1) = 3(16) +(-3)
3f(-2) + f(1) = 48 - 3
3f(-2) + f(1) = 45

Hence the value of the expression 3f(-2) + f(1) is 45

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Step 1: 4 x minus x + 2 + 6 = 6 x + 16
Step 2: 3 x + 8 = 6 x + 16
Step 3: 8 minus 16 = 6 x minus 3 x
Step 4: Negative 8 = 3 x
Step 5: Negative StartFraction 8 Over 3 EndFraction = x

Jorge verifies his solution by substituting Negative StartFraction 8 Over 3 EndFraction into the original equation for x. He determines that his solution is incorrect. Which best describes Jorge’s error?
Jorge distributed incorrectly.
Jorge incorrectly combined like terms.
Jorge incorrectly applied the addition and subtraction properties of equality.
Jorge incorrectly applied the multiplication and division properties of equality

Answers

The error from Jorge's arithmetic operation on the given algebraic expression is that he distributed it incorrectly.

What is an algebraic expression?

An algebraic expression is a mathematical equation that is made up of variables together with arithmetic operations.

From the given expression, we have:

4x - x + 2 + 6 = 6x + 16

Add similar elements together;

3x + 8 = 6x + 16

Using distributive property, subtract 8 from both sides:

3x + 8 - 8 = 6x + 16 - 8

3x = 6x + 8

Simplify

3x-6x = 8

-3x = 8

Divide both sides by -3

-3/-3x = -8/3

x =  -8/3

So from Jorge's calculation, because he distributed incorrectly, we can conclude that could be his error.

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Emily invested $810 in an account paying an interest rate of

Answers

Answer:

complete this

Step-by-step explanation:

yeah do it

my father is 4 times old as me. after 5 years my father will be 3 times old how old is my father now

Answers

Answer:

Step-by-step explanation:

y
8 ⠀⠀⠀⠀
6+€ (1,5)
ATE
E (21)
D (4:1)
-8-6-4-2 2 4 6 8
-6-
Find the area of the triangle.

Answers

The area of the triangle will be 24912 sq. units. Square units and other similar units are used to measure area.

What is the area?

The space filled by a flat form or the surface of an item is known as the area.

The number of unit squares that cover the surface of a closed-form is the figure's area.

For:

(X1, Y1) = (1, 15)

(X2, Y2) = (-2, 1)

d = 14.317821

For:

(X₂, Y₂) = (-2, 1)

(X₃, Y₃) = (4, 5)

d = 7.211103

For applying the pythogorous them we need the right angle triangle obtained by bisect from the mid point.

The value of the base is;

⇒7.2 / 2

⇒3.6

apply the pythogorous theorem for finding the height;

h² = p² + b²

14.31² = p² + 3.6²

p = 13.84

The area of the triangle is;

[tex]\rm A = \frac{1}{2}\times b \times h \\\\ A= \frac{1}{2} \times 3.6 \times 13.84 \\\\ A = 24.912[/tex]

Hence, the area of the triangle will be 24912 sq. units.

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Find the maxima and minima of the following function:
[tex]\displaystyle f(x) = \frac{x^2 - x - 2}{x^2 - 6x + 9}[/tex]

Answers

To find the maxima and minima of the function, we need to calculate the derivative of the function. Note, before the denominator is a perfect square trinomial, so the function can be simplified as

[tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(x) = \frac{x^2 - x - 2}{(x - 3)^2}} \end{gathered}$}[/tex]

So the derivative is:

  [tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(2x - 1)(x - 3)^2 - 2(x - 3)(x^2 - x - 2)}{(x - 3)^4} } \end{gathered}$}[/tex]

Simplifying the numerator, we get:

                 [tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{(x - 3)(-5x + 7)}{(x - 3)^4} = \frac{-5x + 7}{(x - 3)^3} } \end{gathered}$}[/tex]

The function will have a maximum or minimum when f'(x) = 0, that is,

                  [tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f'(x) = \frac{-5x + 7}{(x - 3)^3} = 0 } \end{gathered}$}[/tex]

which is true if -5x + 7 = 0. Then x = 7/5.

To determine whether x = 7/5 is a maximum, we can use the second derivative test or the first derivative test. In this case, it is easier to use the first derivative test to avoid calculating the second derivative. For this, we evaluate f'(x) at a point to the left of x = 7/5 and at a point to the right of it (as long as it is not greater than 3). Since 1 is to the left of 7/5, we evaluate:

                    [tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f(1) = \frac{-5 + 7}{(1 - 3)^3} = \frac{2}{-8} < 0} \end{gathered}$}[/tex]

Likewise, since 2 is to the right of 7/5, then we evaluate:

                                   [tex]\large\displaystyle\text{$\begin{gathered}\sf \displaystyle \bf{\frac{-10 + 7}{(2 - 3)^3} = \frac{-3}{-1} > 0} \end{gathered}$}[/tex]

Note that to the left of 7/5 the derivative is negative (the function decreases) and to the right of 7/5 the derivative is positive (the function increases).

The value of f(x) at 7/5 is:

                               [tex]\large\displaystyle\text{$\begin{gathered}\sf \bf{\displaystyle f\left(\tfrac{7}{5}\right) = \frac{\tfrac{49}{25} - \tfrac{7}{5} - 2}{\tfrac{49}{25} - 6 \cdot \tfrac{7}{5} + 9} = -\frac{9}{16} } \end{gathered}$}[/tex]

This means that [tex]\bf{\left( \frac{7}{5}, -\frac{9}{16} \right)}[/tex] is a minimum (and the only extreme value of f(x)).

[tex]\huge \red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]

Answer:

[tex]\text{Minimum at }\left(\dfrac{7}{5},-\dfrac{9}{16}\right)[/tex]

Step-by-step explanation:

The local maximum and minimum points of a function are stationary points (turning points).  Stationary points occur when the gradient of the function is zero.  Differentiation is an algebraic process that finds the gradient of a curve.

To find the stationary points of a function:

Differentiate f(x)Set f'(x) = 0Solve f'(x) = 0 to find the x-valuesPut the x-values back into the original equation to find the y-values.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]

[tex]\text{Given function}: \quad \text{f}(x)=\dfrac{x^2-x-2}{x^2-6x+9}[/tex]

Differentiate the function using the Quotient Rule:

[tex]\text{Let }u=x^2-x-2 \implies \dfrac{\text{d}u}{\text{d}x}=2x-1[/tex]

[tex]\text{Let }v=x^2-6x+9 \implies \dfrac{\text{d}v}{\text{d}x}=2x-6[/tex]

[tex]\begin{aligned}\implies \dfrac{\text{d}y}{\text{d}x} & =\dfrac{(x^2-6x+9)(2x-1)-(x^2-x-2)(2x-6)}{(x^2-6x+9)^2}\\\\& =\dfrac{(2x^3-13x^2+24x-9)-(2x^3-8x^2+2x+12)}{(x^2-6x+9)^2}\\\\\implies \text{f}\:'(x)& =\dfrac{-5x^2+22x-21}{(x^2-6x+9)^2}\\\\\end{aligned}[/tex]

Set the differentiated function to zero and solve for x:

[tex]\begin{aligned}\implies \text{f}\:'(x)& =0\\\\\implies \dfrac{-5x^2+22x-21}{(x^2-6x+9)^2} & = 0\\\\-5x^2+22x-21 & = 0\\\\-(5x-7)(x-3) & = 0\\\\\implies 5x-7 & = 0 \implies x=\dfrac{7}{5}\\\\\implies x-3 & = 0 \implies x=3\end{aligned}[/tex]

Put the x-values back into the original equation to find the y-values:

[tex]\implies \text{f}\left(\frac{7}{5}\right)=\dfrac{\left(\frac{7}{5}\right)^2-\left(\frac{7}{5}\right)-2}{\left(\frac{7}{5}\right)^2-6\left(\frac{7}{5}\right)+9}=-\dfrac{9}{16}[/tex]

[tex]\implies \text{f}(3)=\dfrac{\left(3\right)^2-\left(3\right)-2}{\left(3\right)^2-6\left(3\right)+9}=\dfrac{4}{0} \implies \text{unde}\text{fined}[/tex]

Therefore, there is a stationary point at:

[tex]\left(\dfrac{7}{5},-\dfrac{9}{16}\right)\:\text{only}[/tex]

To determine if it's a minimum or a maximum, find the second derivative of the function then input the x-value of the stationary point.

If f''(x) > 0 then its a minimum.If f''(x) < 0 then its a maximum.

Differentiate f'(x) using the Quotient Rule:

Simplify f'(x) before differentiating:

[tex]\begin{aligned}\text{f}\:'(x) & =\dfrac{-5x^2+22x-21}{(x^2-6x+9)^2}\\\\& = \dfrac{-(5x-7)(x-3)}{\left((x-3)^2\right)^2}\\\\& = \dfrac{-(5x-7)(x-3)}{(x-3)^4}\\\\& = -\dfrac{(5x-7)}{(x-3)^3}\\\\\end{aligned}[/tex]

[tex]\text{Let }u=-(5x-7) \implies \dfrac{\text{d}u}{\text{d}x}=-5[/tex]

[tex]\text{Let }v=(x-3)^3 \implies \dfrac{\text{d}v}{\text{d}x}=3(x-3)^2[/tex]

[tex]\begin{aligned}\implies \dfrac{\text{d}^2y}{\text{d}x^2} & =\dfrac{-5(x-3)^3+3(5x-7)(x-3)^2}{(x-3)^6}\\\\& =\dfrac{-5(x-3)+3(5x-7)}{(x-3)^4}\\\\\implies \text{f}\:''(x)& =\dfrac{10x-6}{(x-3)^4}\end{aligned}[/tex]

Therefore:

[tex]\text{f}\:''\left(\dfrac{7}{5}\right)=\dfrac{625}{512} > 0 \implies \text{minimum}[/tex]

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