✅ Checkpoint: curvature and convexity

By Better Wise

Quiz

  1. For a twice differentiable scalar function f: R^n -> R, what is the Hessian H(x)?
    • The n x n matrix of second partial derivatives, H_ij = d2f / dx_i dx_j
    • The Jacobian of f itself, which is a 1 x n row vector
    • The n x 1 vector of first partial derivatives
    • The matrix of third partial derivatives, used only for degenerate points
  2. At a critical point x* (so grad f(x*) = 0), the Hessian has eigenvalues 3 and -1. What is x*?
    • Inconclusive, because the eigenvalues have different magnitudes
    • A saddle point
    • A strict local maximum
    • A strict local minimum
  3. In two dimensions, a critical point has det H > 0 and H_11 < 0. What does the determinant shortcut conclude?
    • A saddle point
    • A local minimum
    • The test is inconclusive
    • A local maximum
  4. For f(x, y) = x^2 + y^4, the Hessian at the origin is diag(2, 0). What can you conclude from the second-order test alone?
    • The origin is a strict local minimum
    • The origin is a saddle point
    • Nothing: the test is inconclusive because of the zero eigenvalue
    • The origin is a strict local maximum
  5. Which statements about the Hessian and its eigenvalues are correct? Select all that apply.
    • The second directional derivative along a unit vector v is v^T H v.
    • The Hessian is the Jacobian of the gradient field.
    • A large condition number lambda_max / lambda_min means the curvature is nearly isotropic.
    • The trace of H equals the sum of its eigenvalues.
    • The eigenvalues of H(x) classify x as a minimum or maximum even when grad f(x) is nonzero.
  6. Which statements about Taylor models and convexity are correct? Select all that apply.
    • The quadratic term in the Taylor model is p^T H p, with no factor of one half.
    • The second-order Taylor model is f(x) + grad f(x)^T p + (1/2) p^T H(x) p.
    • H(x) positive semidefinite at a single point is enough to prove f is convex.
    • For a convex differentiable function, every critical point is a global minimum.
    • A twice differentiable f is convex on a convex domain exactly when H(x) is positive semidefinite at every point of that domain.
  7. At a critical point, a Hessian with both positive and negative eigenvalues indicates a ____.
  8. For f(x, y) = x^2 + xy + y^2, the Hessian is [[2, 1], [1, 2]]. Its eigenvalues are 1 and 3, both positive everywhere, so f is strictly ____.
  9. Derive the Hessian of f(x, y) = x^2 y + y^2 + x by hand, then evaluate it at (1, 1) and classify the local behavior of the second-order Taylor model there. Explain what the eigenvalues tell you about the model.
  10. Explain why convexity upgrades a local minimum to a global minimum, and give one example from the lesson where a local minimum is not global because convexity fails.

Sign in to submit answers.