Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. If it is not, enter NONE. F(x, y) = (2x − 4y) i + (−4x + 10y − 7) j f(x, y) = Incorrect: Your answer is incorrect. + K

Step-by-step explanation:

Given that,

F(x, y) = (2x − 4y) i + (−4x + 10y − 7) j

Let P=2x-4y

And Q=-4x+10y+7

For the function to the conservative

Then,

∂P/∂y = ∂Q/∂x

∂P/∂y= -4

∂Q/∂x= -4

Since,

∂P/∂y = ∂Q/∂x

Then, F is conservative

b. Now,

We are looking for a function f so that ∇f = {fx, fy}={2x-4y, -4x+10y-7}

∇f means

∇f= ∂f/∂x •i + ∂f/∂y •j

Then, equating them

∂f/∂x = 2x-4y

∂f=2x-4y ∂x

Integrate both sides

∫ ∂f= ∫2x-4y ∂x

f=2x²/2 - 4xy + f(y)

f(x, y)=x²-4xy +f(y)

∂f/∂y= -4x+f'(y)

Comparing this to

∂f/∂y = −4x + 10y − 7.

Then, f'(y)=10y-7

Integrated both side

f'(y)=10y-7

∂f/∂y=10y-7

∂f=10y-7∂y

∫ ∂f = ∫ 10y-7 ∂y

f(y)= 10y²/2-7y+ C

Here, C=0

f(y)= 5y²-7y

Substituting this into

f(x, y)=x²-4xy +f(y)

f(x, y)=x²-4xy +5y²-7y

Then this is the required function

a

step-by-step explanation:

when a number is between two straight lines it means the absolute value.

to find absolute value you need to know how many numbers is that number away from 0.

when finding absolute value the number will always come out positive

so absolute value of -3 is 3

absolute value of 3/2 is 3/2

absolute value of -2 is 2

absolute value of 3.5 is 3.5

absolute value of -1 is 1

now put these numbers in order:

a)   |-1|, |3/2|, |-2|, |-3|, |3.5|

a - base

b - exponent

so,