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]
2 years ago
9

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

Mathematics
1 answer:
Ray Of Light [21]2 years 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
Find the inner product for (-4,9,8) . (3,2,-2) and state whether the vectors are perpendicular.
drek231 [11]

Answer:

-10; no

Step-by-step explanation:

<h2>-4*3 + 9*2 + 8*-2 = -10</h2><h2></h2><h2>-10 does not equal 0 so it is not perpendicular</h2>

6 0
3 years ago
if you shop in chicago, Illinois, the state sales tax is 9 1/4% what is your total cost when you purchase a $5.00 map of the cit
NNADVOKAT [17]
1.) 1 ÷ 4 =0.25 + 9= 9.25

2.) 9.25 + 5.00 = 14.25
7 0
3 years ago
Find the difference in simplest form. <br> 2b/b^2-2b+1 - 2/b^2-2b+1
kolezko [41]
1) factor the denominators: 2b/(b-1)2 - 2/ (b-1)2
2) MAKE SURE YOU HAVE A COMMON DENOMINATOR
3) combine the numerators: 2b-2/(b-1)2
4) distribute 2 from numerator: 2(b-1)/(b-1)2
5) simplify; answer is 2/(b-1)

7 0
3 years ago
How do i solve this problem
mamaluj [8]
As they are similar corresponding sides are in the same ratio, so

18/15 = x / 4

x = 4*18 / 15
x = 4.8 answer
7 0
2 years ago
Please help me with math please and explain and thanks
lyudmila [28]

Answer:

44

Step-by-step explanation:

there are 4 44s in the data set and that is the most repeated so uts the mode

7 0
2 years ago
Read 2 more answers
Other questions:
  • Lots of points 53 points<br><br> is the √-4 considered a real number why or why not
    13·2 answers
  • Ima houseseller earns a 3% commision for selling a house. if he sells a house for $120000, what is his commision
    11·1 answer
  • Sally has 6 boxes of colored pencils only three of the boxes contain 12 colored pencils each .the remaining boxes each contain 2
    11·1 answer
  • n △ABC, point P∈ AB is so that AP:BP=1:3 and point M is the midpoint of segment CP. Find the area of △ABC if the area of △BMP is
    11·1 answer
  • Twenty people apply for 5 job openings as teacher assistants. How many different groups of people can be hired?
    6·1 answer
  • A runner has a mass of 89 kg he produces aforce of 84 Newtons between the ground and his running shoes How fast does he accelera
    11·1 answer
  • Work out the size of the angle marked X.<br> Give your answer correct to 1 decimal place.
    15·1 answer
  • Pencils come in packages of 10 . Erasers come in package of 12 . Phillip wants to purchase the smallest number of pencils and er
    6·1 answer
  • What is the area of the triangle? ​
    5·2 answers
  • If y = x3 + x,<br> find the value of y when x=-2
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!