Sketch the graph of the relation $x y ^ { 2 } = 2$ using any extrema, intercepts, symmetry, and asymptotes.

Question

Quizplus · Accepted Answer

- Intercepts: 
The x-intercept is obtained by setting y = 0 in the equation xy² = 2 to get x = 0. 
The y-intercept is obtained by setting x = 0 in the equation xy² = 2 to get y = √(2). 
Therefore, there is only one y-intercept at (0, √2) and no x-intercepts.

- Symmetry: 
The relation xy² = 2 is not symmetric with respect to the x-axis, y-axis or origin.

- Asymptotes:
As x approaches infinity or negative infinity, y approaches 0. This means there is a horizontal asymptote at y = 0. However, there are no vertical asymptotes.

- Extrema: 
To find the extrema, we can use the method of Lagrange multipliers with the function f(x,y) = xy^2 and constraint g(x,y) = xy^2 - 2 = 0. 
Solving the system of equations ∇f = λ∇g and g(x,y) = 0. 
We get 4 possible stationary points, (±√2, ∓√2), but none of them are maxima or minima because the function xy^2 does not have an extremum on the curve xy² = 2.

- Sketch:
Putting all of these together, we know that there is one y-intercept at (0, √2), no x-intercepts, no symmetry, a horizontal asymptote at y = 0, and no extrema. 
We can sketch the graph as shown in choice C.

Sketch the Graph of the Relation xy2=2x y ^ { 2 } = 2xy2=2

Sketch the Graph of the Relation $x y ^ { 2 } = 2$