1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Nostrana [21]
1 year ago
9

Find the maximum and minimum values of the curve y=2x³-3x²-12x+10​

Mathematics
1 answer:
Ray Of Light [21]1 year ago
5 0

\underline{  \orange{\huge \boxed{ \frak{Answer : }}}}

Let ,

\sf \large \color{purple} y = 2 {x}^{3}   - 3 {x}^{2}  - 12x + 10  \: --( \: 1 \: )

\: \: \:

Now , Diff wrt ' x ' , we get :

\sf \:  \frac{dy}{dx}  =  \frac{d}{dx} (2 {x}^{3}  - 3 {x}^{2}  - 12x + 10) \\  \sf \: \sf \:  \frac{dy}{dx}  =  \frac{d}{dx} \: 2(3 {x}^{2} ) -  \frac{d}{dx} 3 {x}^{2}  -  \frac{d}{dx} 12x +  \frac{d}{dx} 10 \\  \sf \: \frac{dy}{dx}  =2(3 {x}^{2} ) - 3(2x) - 12(1) + 0 \\  \sf \: \frac{dy}{dx}  =6 {x}^{2}  - 6x - 12 + 0 \\  \: \sf \red{\frac{dy}{dx}  = 6 {x}^{2} 6x - 12 -- (2)}

\: \: \:

For maxima or minima \frac{dy}{dx} = 0

\: \: \:

\sf \: 6 {x}^{2}  - 6x - 12 = 0

\: \: \:

Divided by 6 on both side , we get.

\: \: \:

\sf \:  {x}^{2}  - x - 2 = 0 \\  \sf \:  {x}^{2}  - 2x + x - 2 = 0 \\  \sf \: x(x - 2) + 1(x - 2) = 0 \\  \sf \: (x - 2)(x + 1) = 0 \\  \sf \: x - 2 = 0 \:  \:  \bold or \:  \: x + 1 = 0 \\ \sf \fbox{x = 2 \: }  \: \bold or \:  \fbox{ x =  - 1}

\: \: \:

Again Diff wrt ‘ x ’ , we get.

\sf \: \frac{d}{dx}  =(\frac{dy}{dx}  ) = 6\frac{d}{dx}  - 6\frac{d}{dx}x - \frac{d}{dx}12 \\  \sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  = 6(2x) - 6(1) - 0 \\ \sf \: \sf \bold{  \frac{ {d}^{2}y }{ {dx}^{2} }  =12x - 6}

\: \: \:

At x = 2

\: \: \:

\sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  =12(2) - 6 \\  \:  \:  \:  \sf \:  = 24 - 6 \\   \:  \:  \:  \: \sf \red{  = 18  > 0}

At x = -1

\: \: \:

\sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  =12( - 1) - 6  \\   \:  \:  \: \sf \:   = - 12 - 6 \\  \:  \:  \:  \: \sf \red{  =  - 18 < 0 }

\: \: \:

x = 2 gives minima value of function.

\: \: \:

x = -1 gives maxima value of function.

\: \: \:

Now, put x = 2 in eqⁿ ( 1 )

\: \: \:

\sf \: y \: minima  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: = 2( {2})^{3}  - 3 ({2})^{2}  - 12(2) + 10 \\  \:  \:  \:   \:  \:  \: \:  \:  \:  \:  \:  \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    \:  \:  \:   \sf \:   \:  \:   \:  = 2(8) - 3(4) - 24 + 10 \\ \:  \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    \sf \:   \:  \:  \:  \:  = 16  - 12 - 24 + 10 \\\sf \:   \:  \:  \:  \:  \:  \:  \:  \:  \: =  - 20 + 10 \\\sf \color{red}{\boxed{  =  - 10}}

\: \: \:

<u>The </u><u>Point </u><u>of </u><u>minima </u><u>is </u><u>(</u><u> </u><u>2</u><u> </u><u>,</u><u> </u><u>-</u><u>1</u><u>0</u><u> </u><u>)</u><u>.</u>

\: \: \:

Now , put x = -1 in eqⁿ ( 1 )

\sf \: y \: maxima  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   = 2( { - 1})^{3}  - 3 ({ - 1})^{2}  - 12( - 1) + 10  \\\sf  \color{red}{\boxed{  =  17}}

\: \: \:

<u>The point of maxima value is ( -1 , 17 )</u><u>.</u>

\: \: \:

<u>\:  \:</u>

Hope Helps! :)

You might be interested in
Please answer correctly if you can
tatuchka [14]

Answer:

the answer is 40/100 × 20

6 0
2 years ago
plzzz help smart people, its math will mark branilyest (DONT ANSWER IF YOU DONT KNOW THIS TYPE OF MATH PLZZ SHOW WORK)
sveta [45]

Answer:

6) 15

7)5

8)120 degrees

9)60 degrees

10)9

Step-by-step explanation:

GHIJ is a parallelogram.

Opposite sides of a parallelogram are congruent.

3y - 1 = 2y + 1

3y - 2y = 1 + 1

y = 2

Opposite sides of a parallelogram are congruent.

4x + 3 = x + 12

4x - x = 12 - 3

3x = 9

x = 9/3

x = 3

6)GH = ?

GH = 4x + 3

GH = 4(3) + 3

= 12 + 3

= 15

therefore, GH = 15

7) HI = ?

HI = 2y + 1

= 2(2) + 1

= 4 +1

= 5

therefore, HI = 5

Opposite angles of a parallelogram are equal.

8) m(angle I) = 120 degrees...... (given)

therefore, measure of angle G = measure of angle I

therefore, m(angle G) = 120 degrees

Consecutive angles of a parallelogram are supplementary.

9) m(angle I) + m(angle J) = 180 degrees

120 + m (angle J) = 180

m(angle J) = 180 - 120

= 60 degrees.

The diagonals of a parallelogram bisect each other.

10) JK = 9 .........( given)

JK = HK

therefore, HK = 9

5 0
3 years ago
Which number line best shows the position of square root of 7? Number line with numbers negative 4, negative 3, negative 2, nega
kiruha [24]

Answer:

Number line with numbers negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, 4 labeled. A dot is made at 2.6 and labeled as square root of 7

Step-by-step explanation:

* Lets explain how to solve this problem

- We want to know the position of √7 on the number line, then:

1- Square the √7

# (√7)² = 7

2- Search 7 is between which two square numbers

# 7 is between 4 and 9 which are square numbers

3- ⇒ 4 < 7 < 9 ⇒ take √ for all

# 2 < √7 < 3

∴ The value of √7 is between 2 and 3

* Lets look to the answer to find the correct answer

∵ The first answer is:

   A dot is made at 1.7 and labeled as square root of 7

∵ 1.7 is not between 2 and 3

∴ The first answer is not true

∵ The second answer is:

   A dot is made at 2.6 and labeled as square root of 7

∵ 2.6 is between 2 and 3

∴ The second answer is true

∴ Number line with numbers negative 4, negative 3, negative 2,

  negative 1, 0, 1, 2, 3, 4 labeled. A dot is made at 2.6 and labeled

  as square root of 7

7 0
3 years ago
Read 2 more answers
Lalo has 1,500 minutes per month on his cell phone plan. How many more minutes can he use if he has already talked for 785 minut
Mandarinka [93]
He can use 715 minutes for the rest of the month.
7 0
2 years ago
Read 2 more answers
Multiply.<br>(v+8)(v-8)<br>simplify
san4es73 [151]
V * v = v^2
v * -8 = -8v
8 * v = 8v
8 * -8 = -64

v^2 - 8v + 8v - 64
v^2 - 64
<span>
This was using the distributive property or using the FOIL method</span>
4 0
2 years ago
Other questions:
  • I only need the Evens ;-;
    5·1 answer
  • Round your answers to the nearest hundredth
    11·1 answer
  • An isosceles right triangle has legs that are 10 inches long . how long is the hypotenuse ?
    10·1 answer
  • Find the mean of the data set. –6 3 –2 6 7 –3 –5
    5·1 answer
  • I cannot understand this help me and I do not know what a scale factor is
    8·2 answers
  • What is the value of x?<br> ANSWER ASAP!!
    15·1 answer
  • Please give me the answer i have been working really hard sorry you have to help
    7·1 answer
  • Aaron says that all
    9·1 answer
  • Find the linear equation from Slope of 4; y-intercept of - 5​
    12·1 answer
  • 3<br> x. 3x²+x-3=0<br> Find the zeros
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!