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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?

Thanks ahead of time

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