A similar argument can be expressed for the second envelope theorem which concerns constrained optimization. So, we have an objective function which we have to maximize subject to just one constraint. Once again, alpha is a scalar parameter which belongs to sum interval I. From the start, let's suppose that both functions, f and g are twice continuously differentiable for all values of x and alpha. So, x belongs to Rn space, alpha belongs to I. Let suppose that the Lagrangian function is formed according to the formula. Now, let us write down the first-order conditions for the Lagrangian function. Let's suppose that the solution was found in the form of a critical point and the corresponding lambda value, lambda star. Earlier, we placed an assumption, additional assumption on x star and lambda star also depends on alpha. We said, let's suppose that x star, lambda star belongs to C1 space of functions. Actually, there is no need for such an assumption. Instead, we can think that second order conditions are valid in this case, and let us recall what it means. These second order conditions are applied to the definiteness of the Bordered Hessian metrics. What will be Bordered Hessian metrics in this particular case? Let it be H-tilde. This is n plus one times n plus one matrix made over border. To fill in the entries in the first row, we need to take g and differentiate with respect to the variables. In the same fashion, we fill in the entries in the first column, s for the main block, a square block. Will fill in these entries with the second order derivatives of the Lagrangian function and I can write it using a shorthand, that will be D squared L, so all these variables are with respect to x. Now, let's suppose that the second order condition for the maximization hold, that means that partly, I mean that, all I need to state at the moment, the statement will concern the determinant of H-tilde alone. We know that actually the condition, sufficiency condition for the maximization problem will include another leading principle minus all these H-tilde metrics, but I'm not interested in. So, let's suppose that the sign of the determinant of H-tilde is the same as the sign of minus one to the nth power. That means that this determinant is non-zero. Now, can we apply implicit function theorem to the system of equations? We can, the only thing we need to check will be the non-singularity of the Jacobian matrix. So, let's write down the Jacobian matrix for this particular system of equations and compare with H-tilde, that will produce the result. So in order to form the Jacobian matrix, I'll take the final equation where we differentiate respect to lambda and I'll place it in the, well, above the system, so it will become equation number one. You'll see why because it will resemble H-tilde. So, J will look like that. The l with a lambda being minus J. So, if I differentiate respect to lambda, g doesn't depend on lambda, so I have simply zero. Now, I take this function which is minus g, and I will start differentiating with respect to x, that's how I get minus g1 minus gn. As for the rest of the entries, we start differentiating with respect to x and we get exactly Hessian, here. So, if we compare two matrices H-tilde and J, the only difference is that in geometrics, the first row and first column have minuses in front of first-order derivatives of g. But since we're interested only in determinant of J, the determinant will not change if we multiply by negative one simultaneously, the first row and the first column. So, we get immediately H-tilde and the determinant of H-tilde, as we know from second order conditions is a non-zero number. That means that determinant of J is also a non-zero number, that means the implicit function theorem can be applied. The critical point or the Lagrangian function, and also lambda star, they are continuously differentiable. So we can relax the initial assumption, it will be more fulfilled automatically under the assumption of the second order conditions which it should hold.