Let fx y In x2 y2 Which one of the following curves is a s

Let f(x, y) = In (x^2 + y^2). Which one of the following curves is a simple closed path in the domain of the function f? The unit circle x^2 + y^2 = 1 The rectangle with vertices at (-3, 2), (3, 2, (3, -2), and (-3, -2) The ellipse X^3/a2 + y^2/b^2 = 1

Solution

f(x) = ln (x^2+y^2)

Domain of f(x) is x^2+y^2 >0

a) is not correct as inside the unit circle x=0, y=0 for which f(x) is not defined

b) The rectangle contains (0,0) for which f(x) is not defined

c) The ellipse has (0,0) inside it for which f(x) not defined

d) Here the region inside the curve has all values of (x,y) for which the funciton f(x) is well defined.

So d is right.