A function is given by f(x, y) = 2x^2 - 2y + 3/x^2 - y. Mark the correct option

Question

A function is given by f(x, y) = 2x^2 - 2y + 3/x^2 - y. Mark the correct option in each of the subquestions below. The domain for f consists of all points (x, y) which satisfy x^2 > y y lessthanorequalto x y notequalto x^2 2x - 1 notequalto 0 x^2 + y^2 lessthanorequalto 1 y lessthanorequalto 0 The level curve with equation f(x, y) = 5 can be described as: A parabola with equation y = x^2 - 1. A parabola with equation y = 2x^2 - y + 3. A straight line through (0, 0) with slope 3. A straight line through (0, 3) with slope 2. A circle with center (0, 3/2) and radius 2. A circle with center (1, 2) and radius 1/2.

Explanation / Answer

a) we have given f(x,y)=(2x^2-2y+3)/(x^2-y)

the function takes input values (x,y) and defines the function all input (x,y) values except y=x^2

so the domain of the f(x,y)=(2x^2-2y+3)/(x^2-y) is y not equal to x^2,(x,y) belongs to R

b) we have given the level curve with equation z=f(x,y)=5 and the function f(x,y)=(2x^2-2y+3)/(x^2-y)

solve for y

z=f(x,y)=(2x^2-2y+3)/(x^2-y)

z*(x^2-y)=(2x^2-2y+3)

5x^2-5y=(2x^2-2y+3) since z=f(x,y)=5

5y-2y=5x^2-2x^2-3=3x^2-3

3y=3x^2-3

y=x^2-1

The level curve with equation f(x,y)=5 is A parabola with equation y=x^2-1

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