Assess students' ability to apply the Implicit Function Theorem in Banach spaces to parametrized nonlinear operator equations F(x,λ)=0 by (a) verifying Fréchet differentiability and invertibility of D_xF at a base point, (b) proving local existence, uniqueness and C^1 dependence of solutions x(λ), (c) computing the derivative of the solution map, and (d) checking the theorem's hypotheses on a given example (e.g., a nonlinear integral or PDE operator).