The Jacobian determinant is $e^{2x}(\cos^2y+\sin^2y)=e^{2x}>0$. The inverse function theorem gives a local inverse everywhere. Periodicity in y, f(x,y+2π) = f(x,y), prevents global injectivity. **C**.
More Multivariable Analysis questions
GATE 2025 MA Q21– F:ℝ²→ℝ² is F(x,y)=(x3-3xy2-3x,3x2y-y3-3y). Where is the inverse function theorem…
GATE 2024 MA Q28– For x,y>0, f(x,y)=(x²−y²,2xy). Let g be its local inverse near f(2,1). Find det Dg at…
GATE 2021 MA Q50– The equation xy−z log y+e(xz)=1 defines y=f(x,z) near (x,y,z)=(0,1,1). Find ∇f(0,1).