Test yourself 215) if f''(x)=6x+6 and there is a stationary point at (0,3), find the equation of the curve

Answers

Answer 1

Answer: Ok, so dy/dx=0 at the point (0,3) that is where x=0 and y=3.

$\int { 6x+6dx } \n \n =\frac { 6{ x }^( 2 ) }{ 2 } +6x+C\n \n =3{ x }^( 2 )+6x+C$

$\n \n \therefore \quad { f }^( ' )\left( x \right) =3{ x }^( 2 )+6x+C$

Now, f'(x)=0 when x=0.

Therefore:

$0=C\n \n \therefore \quad { f }^( ' )\left( x \right) =3{ x }^( 2 )+6x$

Now:

$\int { 3{ x }^( 2 ) } +6xdx\n \n =\frac { 3{ x }^( 3 ) }{ 3 } +\frac { 6x^( 2 ) }{ 2 } +C$

$={ x }^( 3 )+3{ x }^( 2 )+C\n \n \therefore \quad f\left( x \right) ={ x }^( 3 )+3{ x }^( 2 )+C$

But when x=0, y=3, therefore:

$3=C\n \n \therefore \quad f\left( x \right) ={ x }^( 3 )+3{ x }^( 2 )+3$

Answer 2

Answer: $f''(x)=6x+6\nf'(x)=\int 6x+6\, dx\nf'(x)=3x^2+6x+C\n\n0=3\cdot0^2+6\cdot0+C\n0=C\nf'(x)=3x^2+6x\n\nf(x)=\int 3x^2+6x\, dx\nf(x)=x^3+3x^2+C\n\n3=0^3+3\cdot0^2+C\n3=C\n\n\boxed{f(x)=x^3+3x^2+3}$

Related Questions

Russian ice breaker yamal can move forward 2.3 meters thick ice at a speed of 5.5 kilometers per hour how long will it take to travel 82.5 kiolometers
4x-8=12 x= How do I find x? And what is it?
The measure of two complementary angles are represented by x+5 and 4x - 15 find the value of x
In a right triangle, if the length of the hypotenuse is 39 and the length of one of the other sides is 15, what is the length of the third side?
-9x+4=-8-5x helppppp plsssss

Find all solutions of the equation. Check your solutions in the original equation. 8 x^3 − 343=0

Answers

$8x^3-343=0\ \ \ \ \ |add\ 343\ to\ both\ sides\n\n8x^3=343\ \ \ \ \ \ |divide\ both\ sides\ by\ 8\n\nx^3=(343)/(8)\n\nx=\sqrt[3]{(343)/(8)}\n\nx=\frac{\sqrt[3]{343}}{\sqrt[3]8}\n\nx=(7)/(2)$

$solution:x=(7)/(2)\n\ncheck:\n\n8\cdot\left((7)/(2)\right)^3-343=0\n\n8\cdot(343)/(8)-343=0\n\n343-343=0\n\n0=0$

$8 x^3-343=0\n\n(2x)^3-7^3=0\ \ \ \ \ \ and\ \ \ \ \ \ a^3-b^3=(a-b)(a^2+ab+b^2)\n\n(2x-7)(4x^2+14x+49)=0\n\n2x-7=0\ \ \ or\ \ \ 4x^2+14x+49=0\n2x=7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta=14^2-4\cdot4\cdot49=196-784=-588\n x= (7)/(2) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Rightarrow\ no\ solution\n.\ \ \ \ \ \ \ \ Ans.\ x= (7)/(2) \n\ncheck:\nL=8x^3-343=\n=8\cdot( (7)/(2) )^3-343=8\cdot (7^3)/(2^3) -343=8\cdot (343)/(8) -343=343-343=0$

Given that (-1,3) is on the graph of f(x), find thecorresponding point for the function
f(x - 1).

Answers

I tried helping but I cant figure it out let me see..

Answer: 0,3

Step-by-step explanation:

A lamp is on sale, and its price is reduced from $80 to $50. What is the percent of decrease? Select from the drop-down menu to correctly complete the statement.

Answers

The percentage decrease is equal to the change in price ( the decreased amount) divided by the original amount, then multiplied by 100. We have,

Percentage decrease = $(80-50)/(80)=0.375*100=37.5$ %

ANSWER: 37.5%

$80 - $50 = $30

$30 * 100 / $ 80 = 37.5%

answer: 37.5% of decrease.

Plz help me I need so much help

Answers

Answer:

see below

Step-by-step explanation:

We have to assume that the question is referring to the doubly-shaded length, the one that is 10 bright blue units long.

You can read the overall length of the shaded area as 5/4, from the markings.

For the next part, you need to count the number of units that are shaded bright blue (10) and the number that are shaded blue overall (15). As a fraction of those shaded blue overall, the ones shaded bright blue are 10/15 = 2/3.

So, the length shaded bright blue is 2/3 of the overall length of 5/4. It is represented by the product ...

$\boxed{(2)/(3)*(5)/(4)}$

Convert 9900 pints into cups

Answers

Answer:19800

Step-by-step explanation:

Answer: 19,800 cups

Step-by-step explanation:

3) Dori created a 2 letter code to get into her waterproof iPad. She only used the letters ABC and D because she was afraid she might forget the combination. Well, Dori forgot anyway and now wants to make a list of all possible combinations so she can get back in A Make a list of all the possible 2 letter codes.

Answers

Answer:

AA, AD, AC, AB (or flipped)

BB, BD, BC, BA (or flipped)

CD, CA, CB, CC (or flipped)

DA, DC, DD, DB (or flipped)

Step-by-step explanation:

Other Questions

What is the total finance charge for a $4,250 loan at 13.25% interest compounded monthly for 24 months? ( I hate asking for help, but I am very bad at math... Could someone possibly help me figure it out? By going through the steps)a. $25.47 b. $202.55 c. $611.20 d. $4,861.20

A function is shown in the table.. . . xg(x). −22. −1−3. 02. 117. . . Which of the following statement is a true for this function? The function is increasing from x = −2 to x = −1.. The function is increasing from x = 0 to x = 1.. The function is decreasing from x = −1 to x = 0.. The function is decreasing from x = 0 to x = 1..

One movie club charges an $100 membership fee and $10 for each movie. Another movie club charges no membership fee and charges $15 per movie. Write and solve an equation to find the number of movies you have to buy for the cost of each club to be the same.

Is (2,4) a solution of the equation y=x-2