# Constrained Optimization: Abstract Problem

I could use some help with these problems.

Suppose we have an objective function $$f(x, y)$$ and a constraint $$y = h(x)$$. Suppose the Lagrangian has a critical point at $$(0, 0, \lambda^*)$$.

1) Explain in a sentence or two how you know that line $$r(t) = (t, th'(0))$$ is tangent to the constraint

2) At the critical point, compute the second derivative of $$f$$ along the line in 1: $$\frac{d^2}{dt^2} f(r(t))|_{t=0}$$ .

3) At the critical point, compute the second derivative of $$f$$ along the graph $$y = h(x)$$: $$\frac{d^2}{dx^2} f(x, h(x))|_{x=0}$$ .

4) Describe the differences between the two computations

For the first part, I'm very confused as to what this line $$r(t)$$ is exactly. I understand that at the critical point, the constraint should be tangent to the objective function. If $$r(t)$$ is tangent to the constraint then it should also be tangent to the objective function I believe, but I'm not sure how to explain this more rigorously.

For the second and third parts I'm also a little confused about how I should be taking the second derivative as the question asks. Can anyone clarify?