Find the particular solution of the differential equation that satisfies the initial condition(s). f ''(x) = 4, f '(2) = 12, f(2) = 18

f(x) = 2*x^2 + 4*x + 2.

Step-by-step explanation:

Ok, we have that:

f''(x) = 4.

Then, integrating, we have that:

f'(x) = 4*x + a.

and from the conditions we know that:

f'(2) = 12 = 4*2 + a = 8 + a

12 = 8 + a

a = 12 - 8 = 4

then f'(x) = 4*x + 4.

now we can integrate again and get:

f(x) = (1/2)*4*x^2 + 4*x + b.

f(x) = 2*x^2 + 4x + b

and we know that f(2) = 18:

f(2) = 18 = 2*2^2 + 4*2 + b = 8 + 8 + b

18 = 16 + b

18 - 16 = b = 2.

Then the function is:

f(x) = 2*x^2 + 4*x + 2.

first option

a function is a relation in which an input is paired with exactly one output.

so, let's find the right answer.

in our first option, our inputs are january, february, and march. the outputs are 30, 28, and 25. each input has it's own output. this is correct!

our second option we know is incorrect, because ms. valverade has two outputs, dogs and cats. this is not a function.

our third option is incorrect as well. the yellow square has two ouputs, the square and the triangle.

lastly, in our fourth option, we see the number 7 two times with a different output. therefore, this is not a function.

remember: each input must be paired with only one output!

a 4/6 = 9/12   not proportional

step-by-step explanation:

to find which are not proportional, we can use cross products

if it is true, they are proportional.   if it is not true, they are not proportional.

a 4/6 = 9/12

4* 12 = 6*9

48 = 54

not this is not proportional  

b 2/3 = 12/18  

2* 18 = 3 * 12

36 = 36

this is proportional

c 5/4 = 10/8  

5*8 = 4*10

40 = 40

this is proportional

d 7/5 = 35/25

7*25 = 5 * 35

175 = 175

is proportional